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Clifford modules

1964-07-01
Michael Atiyah , R. Bott , Alexander Shapiro

On concepts of directional differentiability

1990-09-01
Alexander Shapiro

Study of Exclusive Radiative<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">B</mml:mi></mml:math>Meson Decays

2000-06-05
T. E. Coan , V. Fadeyev , Y. Maravin +7 more
We have studied exclusive, radiative B meson decays to charmless mesons in 9.7x10(6) B&Bmacr; decays accumulated with the CLEO detector. We measure B(B0-->K(*0)(892)gamma) = (4.55(+0.72)(-0. 68)+/-0.34)x10(-5) and B(B+-->K(*+)(892)gamma) = (3.76(+0.89)(-0. … We have studied exclusive, radiative B meson decays to charmless mesons in 9.7x10(6) B&Bmacr; decays accumulated with the CLEO detector. We measure B(B0-->K(*0)(892)gamma) = (4.55(+0.72)(-0. 68)+/-0.34)x10(-5) and B(B+-->K(*+)(892)gamma) = (3.76(+0.89)(-0. 83)+/-0.28)x10(-5). We have searched for CP asymmetry in B-->K(*)(892)gamma decays and measure A(CP) = +0.08+/-0.13+/-0.03. We report the first observation of B-->K(*)(2)(1430)gamma decays with a branching fraction of (1.66(+0.59)(-0.53)+/-0.13)x10(-5). No evidence for the decays B-->rhogamma and B0-->omegagamma is found and we limit B(B-->(rho/omega)gamma)/B(B-->K(*)(892)gamma)<0.32 at 90% C.L.
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Existence and Differentiability of Metric Projections in Hilbert Spaces

1994-02-01
Alexander Shapiro
This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections … This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections always exist, unique and directionally differentiable at boundary points of S. These properties of metric projections are considered for possibly nonconvex sets S. In particular, existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as "nearly convex" sets.

Duality and Optimality Conditions

2000-01-01
Alexander Shapiro , Katya Scheinberg

A cluster realization of $$U_q(\mathfrak {sl}_{\mathfrak {n}})$$ U q ( sl n ) from quantum character varieties

2019-01-19
Gus Schrader , Alexander Shapiro
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Continuous tensor categories from quantum groups I: algebraic aspects

2017-08-27
Gus Schrader , Alexander Shapiro
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under … We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.

Directional differentiability of metric projections onto moving sets at boundary points

1988-05-01
Alexander Shapiro

On differentiability of metric projections in 𝑅ⁿ. I. Boundary case

1987-01-01
Alexander Shapiro
This paper is concerned with metric projections onto a closed subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite-dimensional normed space. Necessary … This paper is concerned with metric projections onto a closed subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite-dimensional normed space. Necessary and in a sense sufficient conditions for directional differentiability of a metric projection at a boundary point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given in terms of approximating cones. It is shown that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined by a number of inequality constraints and a constraint qualification holds, then the approximating cone exists.
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The weight function for the quantum affine algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>)</…

2010-07-01
S. Khoroshkin , Alexander Shapiro
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On $b$-Whittaker functions

2018-06-03
Gus Schrader , Alexander Shapiro
The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a … The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmuller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppalainen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

Rational transformations of algebraic curves and elimination theory

2005-01-01
Alexander Shapiro , Victor Vinnikov
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in … Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve.

Continuous tensor categories from quantum groups I: algebraic aspects

2017-01-01
Gus Schrader , Alexander Shapiro
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under … We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.
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Quantum groups, quantum tori, and the Grothendieck–Springer resolution

2017-10-12
Gus Schrader , Alexander Shapiro
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A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties

2016-01-01
Gus Schrader , Alexander Shapiro
We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured … We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials.

Rational and polynomial representations of Yangians

2014-08-21
S. Khoroshkin , Maxim Nazarov , Alexander Shapiro
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Poisson Geometry of Monic Matrix Polynomials

2015-10-27
Alexander Shapiro
Journal Article Poisson Geometry of Monic Matrix Polynomials Get access Alexander Shapiro Alexander Shapiro 1Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA2Institute of Theoretical & Experimental Physics, … Journal Article Poisson Geometry of Monic Matrix Polynomials Get access Alexander Shapiro Alexander Shapiro 1Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA2Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia Correspondence to be sent to: e-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2016, Issue 17, 2016, Pages 5427–5453, https://doi.org/10.1093/imrn/rnv313 Published: 27 October 2015 Article history Received: 15 May 2014 Revision received: 20 August 2015 Accepted: 06 October 2015 Published: 27 October 2015
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$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

2019-01-01
Gus Schrader , Alexander Shapiro
Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of … Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{\varpi_{i,1},f}$ and $M_{\varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $\mathcal{Q}_\Gamma$ and provide an embedding of the subalgebra $\mathscr{A}'_q$ into the quantized algebra of regular functions on the corresponding cluster variety.
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Quantum decorated character stacks

2021-01-01
David Jordan , Ian Le , Gus Schrader +1 more
We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated … We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated character varieties to encompass also the stacky points, in a way that is both compatible with cutting and gluing and equivariant with respect to canonical actions of the modular group of the surface. In the cases $G=SL_2,PGL_2$ we construct a system of categorical charts and flips on the quantum decorated character stacks which generalize the well--known cluster structures on the Fock--Goncharov moduli spaces.

On Differentiability of Metric Projections in R n , 1: Boundary Case

1987-01-01
Alexander Shapiro

On $b$-Whittaker functions

2018-01-01
Gus Schrader , Alexander Shapiro
The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a … The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmüller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppäläinen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

Quantum groups, quantum tori, and the Grothendieck-Springer resolution

2015-08-28
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the … We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.

Three realizations of the quantum affine algebra U q (A 2 (2)

2010-11-01
Alexander Shapiro
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Several Applications of Bezout Matrices

2006-01-01
Alexander Shapiro , S. Kaplan , Mina Teicher
The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting … The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators, boundaries of quadrature domains etc. We present a survey of several properties of Bezout matrices and their applications in all mentioned topics. We use the framework of Vandermonde vectors because such approach allows us to give new proofs of both classical and modern results and in many cases to obtain new explicit formulas. These explicit formulas can significantly simplify various computational problems and, in particular, make the research of algebraic curves and their applications easier. In addition we wrote a Maple software package, which computes all the formulas. For instance, as Bezout matrices are used in order to compute the image of a rational transformation of an algebraic curve, we used these results to study some connections between small degree rational transformation of an algebraic curve and the braid monodromy of its image.

Rational representations of the Yangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>Y</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="fraktur">g</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">l</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math>

2012-03-13
Alexander Shapiro
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Poisson Geometry of Monic Matrix Polynomials

2014-05-15
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

2019-10-08
Gus Schrader , Alexander Shapiro
Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of … Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{\varpi_{i,1},f}$ and $M_{\varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $\mathcal{Q}_\Gamma$ and provide an embedding of the subalgebra $\mathscr{A}'_q$ into the quantized algebra of regular functions on the corresponding cluster variety.

Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves

2004-01-01
S. Kaplan , Alexander Shapiro , Mina Teicher
We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such … We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is given along with full classification of the images of two intersecting lines under degree 2 rational transformation. We also establish a connection between degree 2 rational transformations and the local braid monodromy of the image at the intersecting point of two lines. Moreover, we present an example of two birationally isomorphic curves with the same braid monodromy type and non diffeomorphic real parts.

Rational transformations of systems of commuting nonselfadjoint operators

2005-01-01
Alexander Shapiro , Victor Vinnikov
The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation … The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation of algebraic curves, we define the notion of a rational transformation of a system of commuting nonselfadjoint operators.

Dual pairs of quantum moment maps and doubles of Hopf algebras

2015-10-18
Gus Schrader , Alexander Shapiro
For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism … For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.

Три реализации квантовой аффинной алгебры $U_q(A_2^{(2)})$

2010-01-01
Александр Михайлович Шапиро , Alexander Shapiro
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A cluster realization of the $U_q(\mathfrak{sl}_n)$ $R$-matrix from quantum character varieties

2016-07-01
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum torus algebra $\mathfrak{D}_n$. The algebra $\mathfrak{D}_n$ arises as a quantum cluster chart on a certain quantum character … We construct an algebra embedding of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum torus algebra $\mathfrak{D}_n$. The algebra $\mathfrak{D}_n$ arises as a quantum cluster chart on a certain quantum character variety associated to the (decorated) punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of a quantum character variety associated to the (decorated) twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a (refined) factorization of the $R$-matrix into quantum dilogarithms of cluster variables.

Symplectic leaves for rational loops with Yangian $r$-matrix

2014-05-15
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

Dual pairs of quantum moment maps and doubles of Hopf algebras

2017-09-05
Gus Schrader , Alexander Shapiro
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Preliminary Results on |V_ub| from Inclusive Semileptonic B Decays with Neutrino Reconstruction

2002-01-01
A. Bornheim , D. L. Hartill , C. J. Stepaniak +7 more
We present an analysis of the composition of inclusive semileptonic B meson decays using 9.4/fb of e+ e- data taken with the CLEO detector at the Upsilon(4S) resonance. In addition … We present an analysis of the composition of inclusive semileptonic B meson decays using 9.4/fb of e+ e- data taken with the CLEO detector at the Upsilon(4S) resonance. In addition to measuring the charged lepton kinematics, the neutrino 4-vector is inferred using the hermeticity of the detector. We perform a maximum likelihood fit over the full three-dimensional differential decay distribution for the fractional contributions from the B -&gt; Xc l nu processes with Xc = D, D*, D**, and non-resonant Xc, and the process B -&gt; Xu l nu. From the fit results we extract |V_ub|= (4.05 +- 0.18 +- 0.58 +- 0.25 +- 0.21 +-0.56) x 10^{-3} where the errors are statistical, detector systematics, B -&gt; Xc l nu model dependence, B -&gt; Xu l nu model dependence, and theoretical uncertainty respectively.

Score Functions

2013-01-01
Reuven Y. Rubinstein , Alexander Shapiro , Stan Uryasev

Clifford Modules

1994-01-01
Michael Atiyah , R. Bott , Alexander Shapiro

Dual pairs of quantum moment maps and doubles of Hopf algebras

2015-01-01
Gus Schrader , Alexander Shapiro
For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism … For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.
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Poisson Geometry of Monic Matrix Polynomials

2014-01-01
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.
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Quantum groups, quantum tori, and the Grothendieck-Springer resolution

2015-01-01
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the … We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.
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Ruijsenaars wavefunctions as modular group matrix coefficients

2024-11-27
Philippe Di Francesco , Rinat Kedem , S. Khoroshkin +2 more
Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $$S\in SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> … Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $$S\in SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> acting on the Hilbert space of GL (2) quantum Teichmüller theory on the punctured torus. The GL (2) Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed GL (2)-local systems on the punctured torus, and an $$SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -equivariant embedding of the GL (2) spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety.
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Cluster structure on genus 2 spherical DAHA: seven-colored flower

2024-02-27
S. Arthamonov , Leonid Chekhov , Philippe Di Francesco +4 more
We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The … We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.
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Ruijsenaars wavefunctions as modular group matrix coefficients

2024-11-27
Philippe Di Francesco , Rinat Kedem , S. Khoroshkin +2 more
Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $$S\in SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> … Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $$S\in SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> acting on the Hilbert space of GL (2) quantum Teichmüller theory on the punctured torus. The GL (2) Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed GL (2)-local systems on the punctured torus, and an $$SL(2,{\mathbb {Z}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -equivariant embedding of the GL (2) spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety.
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Cluster structure on genus 2 spherical DAHA: seven-colored flower

2024-02-27
S. Arthamonov , Leonid Chekhov , Philippe Di Francesco +4 more
We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The … We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.
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Quantum decorated character stacks

2021-01-01
David Jordan , Ian Le , Gus Schrader +1 more
We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated … We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated character varieties to encompass also the stacky points, in a way that is both compatible with cutting and gluing and equivariant with respect to canonical actions of the modular group of the surface. In the cases $G=SL_2,PGL_2$ we construct a system of categorical charts and flips on the quantum decorated character stacks which generalize the well--known cluster structures on the Fock--Goncharov moduli spaces.

$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

2019-10-08
Gus Schrader , Alexander Shapiro
Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of … Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{\varpi_{i,1},f}$ and $M_{\varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $\mathcal{Q}_\Gamma$ and provide an embedding of the subalgebra $\mathscr{A}'_q$ into the quantized algebra of regular functions on the corresponding cluster variety.

A cluster realization of $$U_q(\mathfrak {sl}_{\mathfrak {n}})$$ U q ( sl n ) from quantum character varieties

2019-01-19
Gus Schrader , Alexander Shapiro
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$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

2019-01-01
Gus Schrader , Alexander Shapiro
Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of … Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q \subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{\varpi_{i,1},f}$ and $M_{\varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $\mathcal{Q}_\Gamma$ and provide an embedding of the subalgebra $\mathscr{A}'_q$ into the quantized algebra of regular functions on the corresponding cluster variety.
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On $b$-Whittaker functions

2018-06-03
Gus Schrader , Alexander Shapiro
The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a … The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmuller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppalainen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

On $b$-Whittaker functions

2018-01-01
Gus Schrader , Alexander Shapiro
The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a … The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmüller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppäläinen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

Quantum groups, quantum tori, and the Grothendieck–Springer resolution

2017-10-12
Gus Schrader , Alexander Shapiro
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Dual pairs of quantum moment maps and doubles of Hopf algebras

2017-09-05
Gus Schrader , Alexander Shapiro
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Continuous tensor categories from quantum groups I: algebraic aspects

2017-08-27
Gus Schrader , Alexander Shapiro
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under … We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.

Continuous tensor categories from quantum groups I: algebraic aspects

2017-01-01
Gus Schrader , Alexander Shapiro
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under … We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.
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A cluster realization of the $U_q(\mathfrak{sl}_n)$ $R$-matrix from quantum character varieties

2016-07-01
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum torus algebra $\mathfrak{D}_n$. The algebra $\mathfrak{D}_n$ arises as a quantum cluster chart on a certain quantum character … We construct an algebra embedding of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum torus algebra $\mathfrak{D}_n$. The algebra $\mathfrak{D}_n$ arises as a quantum cluster chart on a certain quantum character variety associated to the (decorated) punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of a quantum character variety associated to the (decorated) twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a (refined) factorization of the $R$-matrix into quantum dilogarithms of cluster variables.

A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties

2016-01-01
Gus Schrader , Alexander Shapiro
We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured … We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials.

Poisson Geometry of Monic Matrix Polynomials

2015-10-27
Alexander Shapiro
Journal Article Poisson Geometry of Monic Matrix Polynomials Get access Alexander Shapiro Alexander Shapiro 1Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA2Institute of Theoretical & Experimental Physics, … Journal Article Poisson Geometry of Monic Matrix Polynomials Get access Alexander Shapiro Alexander Shapiro 1Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA2Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia Correspondence to be sent to: e-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2016, Issue 17, 2016, Pages 5427–5453, https://doi.org/10.1093/imrn/rnv313 Published: 27 October 2015 Article history Received: 15 May 2014 Revision received: 20 August 2015 Accepted: 06 October 2015 Published: 27 October 2015
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Dual pairs of quantum moment maps and doubles of Hopf algebras

2015-10-18
Gus Schrader , Alexander Shapiro
For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism … For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.

Quantum groups, quantum tori, and the Grothendieck-Springer resolution

2015-08-28
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the … We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.

Dual pairs of quantum moment maps and doubles of Hopf algebras

2015-01-01
Gus Schrader , Alexander Shapiro
For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism … For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.
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Quantum groups, quantum tori, and the Grothendieck-Springer resolution

2015-01-01
Gus Schrader , Alexander Shapiro
We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the … We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.
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Rational and polynomial representations of Yangians

2014-08-21
S. Khoroshkin , Maxim Nazarov , Alexander Shapiro
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Symplectic leaves for rational loops with Yangian $r$-matrix

2014-05-15
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

Poisson Geometry of Monic Matrix Polynomials

2014-05-15
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

Poisson Geometry of Monic Matrix Polynomials

2014-01-01
Alexander Shapiro
We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all … We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.
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Score Functions

2013-01-01
Reuven Y. Rubinstein , Alexander Shapiro , Stan Uryasev

Rational representations of the Yangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>Y</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="fraktur">g</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">l</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math>

2012-03-13
Alexander Shapiro
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Three realizations of the quantum affine algebra U q (A 2 (2)

2010-11-01
Alexander Shapiro
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The weight function for the quantum affine algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>)</…

2010-07-01
S. Khoroshkin , Alexander Shapiro
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Три реализации квантовой аффинной алгебры $U_q(A_2^{(2)})$

2010-01-01
Александр Михайлович Шапиро , Alexander Shapiro
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Several Applications of Bezout Matrices

2006-01-01
Alexander Shapiro , S. Kaplan , Mina Teicher
The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting … The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators, boundaries of quadrature domains etc. We present a survey of several properties of Bezout matrices and their applications in all mentioned topics. We use the framework of Vandermonde vectors because such approach allows us to give new proofs of both classical and modern results and in many cases to obtain new explicit formulas. These explicit formulas can significantly simplify various computational problems and, in particular, make the research of algebraic curves and their applications easier. In addition we wrote a Maple software package, which computes all the formulas. For instance, as Bezout matrices are used in order to compute the image of a rational transformation of an algebraic curve, we used these results to study some connections between small degree rational transformation of an algebraic curve and the braid monodromy of its image.

Rational transformations of algebraic curves and elimination theory

2005-01-01
Alexander Shapiro , Victor Vinnikov
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in … Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve.

Rational transformations of systems of commuting nonselfadjoint operators

2005-01-01
Alexander Shapiro , Victor Vinnikov
The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation … The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation of algebraic curves, we define the notion of a rational transformation of a system of commuting nonselfadjoint operators.

Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves

2004-01-01
S. Kaplan , Alexander Shapiro , Mina Teicher
We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such … We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is given along with full classification of the images of two intersecting lines under degree 2 rational transformation. We also establish a connection between degree 2 rational transformations and the local braid monodromy of the image at the intersecting point of two lines. Moreover, we present an example of two birationally isomorphic curves with the same braid monodromy type and non diffeomorphic real parts.

Preliminary Results on |V_ub| from Inclusive Semileptonic B Decays with Neutrino Reconstruction

2002-01-01
A. Bornheim , D. L. Hartill , C. J. Stepaniak +7 more
We present an analysis of the composition of inclusive semileptonic B meson decays using 9.4/fb of e+ e- data taken with the CLEO detector at the Upsilon(4S) resonance. In addition … We present an analysis of the composition of inclusive semileptonic B meson decays using 9.4/fb of e+ e- data taken with the CLEO detector at the Upsilon(4S) resonance. In addition to measuring the charged lepton kinematics, the neutrino 4-vector is inferred using the hermeticity of the detector. We perform a maximum likelihood fit over the full three-dimensional differential decay distribution for the fractional contributions from the B -&gt; Xc l nu processes with Xc = D, D*, D**, and non-resonant Xc, and the process B -&gt; Xu l nu. From the fit results we extract |V_ub|= (4.05 +- 0.18 +- 0.58 +- 0.25 +- 0.21 +-0.56) x 10^{-3} where the errors are statistical, detector systematics, B -&gt; Xc l nu model dependence, B -&gt; Xu l nu model dependence, and theoretical uncertainty respectively.

Study of Exclusive Radiative<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">B</mml:mi></mml:math>Meson Decays

2000-06-05
T. E. Coan , V. Fadeyev , Y. Maravin +7 more
We have studied exclusive, radiative B meson decays to charmless mesons in 9.7x10(6) B&Bmacr; decays accumulated with the CLEO detector. We measure B(B0-->K(*0)(892)gamma) = (4.55(+0.72)(-0. 68)+/-0.34)x10(-5) and B(B+-->K(*+)(892)gamma) = (3.76(+0.89)(-0. … We have studied exclusive, radiative B meson decays to charmless mesons in 9.7x10(6) B&Bmacr; decays accumulated with the CLEO detector. We measure B(B0-->K(*0)(892)gamma) = (4.55(+0.72)(-0. 68)+/-0.34)x10(-5) and B(B+-->K(*+)(892)gamma) = (3.76(+0.89)(-0. 83)+/-0.28)x10(-5). We have searched for CP asymmetry in B-->K(*)(892)gamma decays and measure A(CP) = +0.08+/-0.13+/-0.03. We report the first observation of B-->K(*)(2)(1430)gamma decays with a branching fraction of (1.66(+0.59)(-0.53)+/-0.13)x10(-5). No evidence for the decays B-->rhogamma and B0-->omegagamma is found and we limit B(B-->(rho/omega)gamma)/B(B-->K(*)(892)gamma)<0.32 at 90% C.L.
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Duality and Optimality Conditions

2000-01-01
Alexander Shapiro , Katya Scheinberg

Existence and Differentiability of Metric Projections in Hilbert Spaces

1994-02-01
Alexander Shapiro
This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections … This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections always exist, unique and directionally differentiable at boundary points of S. These properties of metric projections are considered for possibly nonconvex sets S. In particular, existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as "nearly convex" sets.

Clifford Modules

1994-01-01
Michael Atiyah , R. Bott , Alexander Shapiro

On concepts of directional differentiability

1990-09-01
Alexander Shapiro

Directional differentiability of metric projections onto moving sets at boundary points

1988-05-01
Alexander Shapiro

On differentiability of metric projections in 𝑅ⁿ. I. Boundary case

1987-01-01
Alexander Shapiro
This paper is concerned with metric projections onto a closed subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite-dimensional normed space. Necessary … This paper is concerned with metric projections onto a closed subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite-dimensional normed space. Necessary and in a sense sufficient conditions for directional differentiability of a metric projection at a boundary point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given in terms of approximating cones. It is shown that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined by a number of inequality constraints and a constraint qualification holds, then the approximating cone exists.
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On Differentiability of Metric Projections in R n , 1: Boundary Case

1987-01-01
Alexander Shapiro

Clifford modules

1964-07-01
Michael Atiyah , R. Bott , Alexander Shapiro
Coauthor Papers Together
Gus Schrader 18
D. Cronin-Hennessy 2
H. Schwarthoff 2
B. Gittelman 2
T. K. Pedlar 2
T. Hart 2
L. Gibbons 2
S. Khoroshkin 2
D. Besson 2
Mina Teicher 2
C. Plager 2
J. R. Patterson 2
E. Nordberg 2
R. Bott 2
Michael Atiyah 2
C. Sedlack 2
Philippe Di Francesco 2
A. L. Lyon 2
R. A. Briere 2
C. Gwon 2
W. Sun 2
V. Savinov 2
H. Severini 2
I. Dankó 2
M. M. Zoeller 2
G. Brandenburg 2
Rinat Kedem 2
J. B. Thayer 2
M. Selen 2
M. Saleem 2
S. Stone 2
G. H. A. Viehhauser 2
T. E. Coan 2
M. Artuso 2
K. M. Ecklund 2
B. K. Heltsley 2
A. Magerkurth 2
R. Poling 2
S. Kaplan 2
J. E. Duboscq 2
K. Berkelman 2
A. J. Weinstein 2
D. L. Hartill 2
K. Bukin 2
D. Riley 2
E. Dambasuren 2
R. Mahapatra 2
A. Warburton 2
Victor Vinnikov 2
D. L. Kreinick 2

The quantum dilogarithm and representations of quantum cluster varieties

2008-09-11
V. V. Fock , A. B. Goncharov
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Positive Representations of Split Real Quantum Groups and Future Perspectives

2013-01-11
Igor Frenkel , Ivan Chi-Ho Ip
Journal Article Positive Representations of Split Real Quantum Groups and Future Perspectives Get access Igor B. Frenkel, Igor B. Frenkel 1Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, … Journal Article Positive Representations of Split Real Quantum Groups and Future Perspectives Get access Igor B. Frenkel, Igor B. Frenkel 1Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520, USA Search for other works by this author on: Oxford Academic Google Scholar Ivan C. H. Ip Ivan C. H. Ip 2Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2014, Issue 8, 2014, Pages 2126–2164, https://doi.org/10.1093/imrn/rns288 Published: 11 January 2013 Article history Received: 21 September 2012 Accepted: 03 December 2012 Published: 11 January 2013
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Moduli spaces of local systems and higher Teichmüller theory

2006-06-01
V. V. Fock , A. B. Goncharov
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Quantum cluster algebras

2004-09-30
Arkady Berenstein , Andrei Zelevinsky
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Dressing Transformations and Poisson Group Actions

1985-12-31
Michael Semenov-Tian-Shansky
Poisson properties of dressing transformations in soliton theory are explained. Poisson properties of dressing transformations in soliton theory are explained.
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Drinfeld double of GLn and generalized cluster structures

2017-11-28
Misha Gekhtman , Michael Shapiro , Alek Vainshtein
We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on … We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the Poisson bracket on the dual Poisson--Lie group.
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Group-like elements in quantum groups, and Feigin's conjecture

1996-01-01
Arkady Berenstein
Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf … Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion (with respect to the natural grading) of the quantized enveloping algebra of ${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let $P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k

On the discriminant function of two commuting nonselfadjoint operators

1980-03-01
N. Kravitsky

Self-adjoint determinantal representations of real plane curves

1993-12-01
Victor Vinnikov

Positive Representations of Split Real Simply-Laced Quantum Groups

2020-06-17
Ivan Chi-Ho Ip
We construct the positive principal series representations for \mathcal U_q(\mathfrak g_\mathbb R) where \mathfrak g is of simply-laced type, parametrized by \mathbb R_{\geq 0}^r where r is the rank of … We construct the positive principal series representations for \mathcal U_q(\mathfrak g_\mathbb R) where \mathfrak g is of simply-laced type, parametrized by \mathbb R_{\geq 0}^r where r is the rank of \mathfrak g . We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group \mathbf U_{\mathfrak q \tilde{\mathfrak q}}(\mathfrak g_\mathbb R) of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.
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On the Drinfeld double and the Heisenberg double of a Hopf algebra

1994-06-01
Jiang-Hua Lu

Positive representations of non-simply-laced split real quantum groups

2014-12-22
Ivan Chi-Ho Ip
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Poisson Lie groups, dressing transformations, and Bruhat decompositions

1990-01-01
Jiang-Hua Lu , Alan Weinstein
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Kac-Moody Groups, their Flag Varieties and Representation Theory

2002-01-01
Shrawan Kumar
Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge bras which we refer to as the finite … Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge bras which we refer to as the finite case. The t

On quasidifferentiable mappings

1983-01-01
V. F. Demyanov , A. M. Rubinov
Abstract In nondifferentiable optimization an important role is performed by a sub-differential — a set of linear functionals — which, in one sense or another, locally approximate a given function. … Abstract In nondifferentiable optimization an important role is performed by a sub-differential — a set of linear functionals — which, in one sense or another, locally approximate a given function. For a convex function the subdifferential enables one to describe necessary conditions for a minimum, to compute directional derivative, to find steepest descent directions. That's why in any attempts have been made to extend the concept of subdifferential to nonconvex nonsmooth functions. It has been shown by the authors that for a rather large family of functions it is useful and natural to consider as an approximating tool not a single set (the subdifferential), but a pair of sets (the quasidifferential). The family of quasidifferential functions is a linear space closed with respect to all algebraic operations as well as the operations of taking pointwise maximum and minimum. In this paper a short survey of some properties of quasidifferentiable functions is presented. The notion of quasidifferentiable mappings is introduced. The quasidifferentiability of a composition.of quasidifferentiable mappings is proved. Some applications of the composition theorem are discussed.

The Beilinson-Bernstein correspondence for quantized enveloping algebras

2005-01-07
Toshiyuki Tanisaki
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Weight Functions and Drinfeld Currents

2007-10-05
B. Enriquez , S. Khoroshkin , S. Pakuliak
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Weyl group extension of quantized current algebras

2000-03-01
Jíntai Ding , S. Khoroshkin
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Separation of variables in the classical integrableSL(3) magnetic chain

1992-11-01
E. K. Sklyanin
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Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, I

2016-01-01
Hiraku Nakajima
Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact … Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-K\"ahler manifold, such as instanton moduli spaces on $\mathbb R^4$, $SU(2)$-monopole moduli spaces on $\mathbb R^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $\sigma$-model on the $2$-sphere associated with $(G,\mathbf M)$. In this first part, we check that the cohomology group has the correct graded dimensions expected from the monopole formula proposed by Cremonesi, Hanany and Zaffaroni arXiv:1309.2657. A ring structure (on the cohomology of a modified moduli space) will be introduced in the sequel of this paper.
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Cluster structures on quantum coordinate rings

2012-07-22
Christof Geiß , Bernard Leclerc , Jan Schröer
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Algebraic bethe ansatz for the Izergin-Korepin R matrix

1988-08-01
V. Tarasov

Clebsch–Gordan and Racah–Wigner Coefficients for a Continuous Series of Representations of ?q (??(2, ℝ))

2001-12-01
Bénédicte Ponsot , J. Teschner
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Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras

2019-01-01
Michael Finkelberg , Alexander Tsymbaliuk
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Rational transformations of algebraic curves and elimination theory

2005-01-01
Alexander Shapiro , Victor Vinnikov
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in … Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve.

Continuous tensor categories from quantum groups I: algebraic aspects

2017-08-27
Gus Schrader , Alexander Shapiro
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under … We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.

The Feigin Tetrahedron

2015-03-19
Dylan Rupel
The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras.More precisely, we define generalized Feigin homomorphisms from a quantum shuffle … The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras.More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra.In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined.To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms.These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra.The second goal in this project is to better understand the dual canonical basis conjecture for skewsymmetrizable quantum cluster algebras.In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras.We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra.Indeed, the main conjecture of this note would establish this "KLR conjecture" for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras.We sketch a proof in the symmetric case giving an alternative to the proof of Kimura-Qin that all non-initial cluster variables in an acyclic skew-symmetric quantum cluster algebra are contained in the dual canonical basis.With these results in mind we interpret the cluster mutations directly in terms of the representation theory of the KLR algebra.
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q-Weyl group and a multiplicative formula for universalR-matrices

1990-11-01
A. N. Kirillov , Nicolai Reshetikhin
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The uniqueness theorem for the universal R-matrix

1992-03-01
S. Khoroshkin , V.N. Tolstoy

Lectures on Quantum Groups

1995-11-15
Jens Carsten Jantzen
Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations … Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations of $U_q(\mathfrak g)$ Examples of representations The center and bilinear forms $R$-matrices and $k_q[G]$ Braid group actions and PBW type basis Proof of proposition 8.28 Crystal bases I Crystal bases II Crystal bases III References Notations Index.

Modular Double of Quantum Group

1999-01-01
Ludvig Faddeev
An extension of Quantum Group is described. We propose to unite the quantum groups with parameter q and with parameter modularly dual to q. An extension of Quantum Group is described. We propose to unite the quantum groups with parameter q and with parameter modularly dual to q.

Liouville bootstrap via harmonic analysis on a noncompact quantum group

1999-01-01
Bénédicte Ponsot , J. Teschner
The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence … The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c&gt;25. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1

The various definitions of the derivative in linear topological spaces

1968-08-31
V. I. Averbukh , O. G. Smolyanov
The object of this article is to give a survey of the existing definitions of the operation of differentiation in linear topological spaces (l.t.s.) and to show the connections between … The object of this article is to give a survey of the existing definitions of the operation of differentiation in linear topological spaces (l.t.s.) and to show the connections between them. There are at present more than a score of definitions of the derivative of a map of one l.t.s. into another. These definitions are stated in what are superficially completely different ways, and the authors who proposed new definitions have not as a rule concerned themselves with the relations between the new definitions and those already known. This leads to an impression of chaos. The two existing surveys of the theory of differentiation in l.t.s. do not alter this impression. The first, due to Hyers, is a part of an article on l.t.s. which appeared in 1945 and is out of date. The second, due to Keller, was published in 1964 but is not complete. In the first place, it is far from giving all the definitions known at that time; in the second place, it considers only locally convex l.t.s; and in the third place, the connections between the definitions discussed are not described fully enough. It turns out - and this is really remarkable - that in fact all the definitions for the derivative that have been proposed up to now can be grouped into a small number of classes, each consisting of equivalent definitions, and that there is a simple scheme of mutual relations between these classes. The present article follows on our article [1]. However, the latter had a different object: to give a systematic account of differential calculus in l.t.s. The account was based on one definition of the derivative, which seemed to us the most successful. (In any case many results do not depend on what definition is taken as basis.) A short survey of some definitions of the derivative was given in the introduction to [1], and a table of connections between them was given without proofs. Thus, this article and [1] complement one another and together constitute a united whole. An idea of the contents of the article can be obtained from the chapter headings. We make a few remarks. We begin with a historical sketch of the development of the idea of the derivative from 1887 to the present. Then follows a list of all definitions that have been proposed (Table 1). Next, this list is divided into classes of mutually equivalent definitions. The enumeration of inequivalent definitions is in Table 2. The connections between them are shown in a diagram. Table 3 gives an enumeration of counterexamples, which demonstrate that no implications are valid except those in the diagram. All the l.t.s. discussed are assumed to be separated l.t.s. on the field R of real numbers. Apart from the contents of the elementary university courses, all that is needed for the understanding of this article is a knowledge of the rudimentary facts of the theory of l.t.s., for example those in Chapter 9 of the book by Kantorovich and Akilov (1959).

Yangians and Classical Lie Algebras

2007-10-30
Alexander Molev
Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction … Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Kirillov-Reshetikhin Conjecture: The General Case

2009-08-12
D. Paredes Hernandez
We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and … We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and we get explicit formulas for the character of their tensor products (the untwisted case was treated in [16, 33, 34]). The proof provides several new developments for the representation theory of twisted quantum affine algebras, in particular on the twisted Frenkel–Reshetikhin q-characters that we define (expected in [11, 12]) and on the parameterization of simple finite dimensional representations without Drinfeld presentation. We also prove the twisted T-system [30]. As an application, we get a proof of explicit (q)-characters formulas conjectured in various papers. We prove an isomorphism of Grothendieck rings between a twisted quantum affine algebra and the corresponding simply-laced quantum affine algebra.
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Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems

1976-09-01
Stephen M. Robinson
This paper develops a condition for stability of the solution set of a system of nonlinear inequalities over a closed convex set in a Banach space, when the functions defining … This paper develops a condition for stability of the solution set of a system of nonlinear inequalities over a closed convex set in a Banach space, when the functions defining the inequalities are subjected to small perturbations. The condition involves the linearization of the system about a point; it is shown to be sufficient and, under a weak additional hypothesis, also necessary for stability. Quantitative estimates for the changes in the solution set are obtained.

Lie group valued moment maps

1998-01-01
Anton Alekseev , Anton Malkin , Eckhard Meinrenken
We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra.The theory includes … We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra.The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map.As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G X • • • X G .
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The weight function for the quantum affine algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>)</…

2010-07-01
S. Khoroshkin , Alexander Shapiro
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Parametrizations of Canonical Bases and Totally Positive Matrices

1996-09-01
Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

Lectures on quantum groups

2002-01-01
Pavel Etingof , Olivier Schiffmann
Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. Covered topics include: Poisson algebras and quantization, Poisson-Lie groups, coboundary Lie … Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. Covered topics include: Poisson algebras and quantization, Poisson-Lie groups, coboundary Lie bialgebras, Drinfelds double construction, Belavin-Drinfeld classification, Infinite dimensional Lie bialgebras, Hopf algebras, Quantized universal enveloping algebras, formal groups and h-formal groups, infinite dimensional quantum groups, the quantum double, tensor categories and quasi Hopf-algebras, braided tensor categories, KZ equations and the Drinfeld Category, Quasi-Hpf enveloping algebras, Lie associators, Fiber functors and Tannaka-Driein duality, Quantization of finite Lie bialgebras, Universal constructions, Universal quantization, Dequantization and the equivalence theorem, KZ associator and multiple zeta functions, and Mondoromy of trigonometric KZ equations. Probems are given with each subject and an answer key is included. New paperback re-issue of the revised second edition.

Degenerate affine hecke algebras and Yangians

1986-01-01
Vladimir Drinfeld

Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras

1998-01-01
Ivan Cherednik
<!-- *** Custom HTML *** --> This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to … <!-- *** Custom HTML *** --> This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to follow closely the notes of the lectures not yielding to the temptation of giving more examples and names. The focus is on the relations of the Knizhnik-Zamolodchikov equations and Kac-Moody algebras to a new theory of spherical and hypergeometric functions based on affine and double affine Hecke algebras. Here mathematics and physics are closer than Siamese twins. We did not try to separate them, but the course turned out to be mainly about the mathematical issues. However we hope that the paper will be understandable for both physicists and mathematicians, for those who want to master the new Hecke algebra technique.
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Mickelsson algebras and representations of Yangians

2011-10-12
S. Khoroshkin , Maxim Nazarov
Let $\operatorname {Y}(\mathfrak {gl}_n)$ be the Yangian of the general linear Lie algebra $\mathfrak {gl}_n$. We denote by $\operatorname {Y}(\mathfrak {sp}_n)$ and $\operatorname {Y}(\mathfrak {so}_n)$ the twisted Yangians corresponding to … Let $\operatorname {Y}(\mathfrak {gl}_n)$ be the Yangian of the general linear Lie algebra $\mathfrak {gl}_n$. We denote by $\operatorname {Y}(\mathfrak {sp}_n)$ and $\operatorname {Y}(\mathfrak {so}_n)$ the twisted Yangians corresponding to the symplectic and orthogonal subalgebras in the Lie algebra $\mathfrak {gl}_n$. These twisted Yangians are one-sided coideal subalgebras in the Hopf algebra $\operatorname {Y}(\mathfrak {gl}_n)$. We provide realizations of irreducible modules of the algebras $\operatorname {Y}(\mathfrak {sp}_n)$ and $\operatorname {Y}(\mathfrak {so}_n)$ as certain quotients of tensor products of symmetic and exterior powers of the vector space $\mathbb {C}^n$. For the Yangian $\operatorname {Y} (\mathfrak {gl}_n)$ such realizations have been known, but we give new proofs of these results. For the twisted Yangian $\operatorname {Y}(\mathfrak {sp}_n)$, we realize all irreducible finite-dimensional modules. For the twisted Yangian $\operatorname {Y}(\mathfrak {so}_n)$, we realize all those irreducible finite-dimensional modules, where the action of the Lie algebra $\mathfrak {so}_n$ integrates to an action of the special orthogonal Lie group $\mathrm {SO}_n$. Our results are based on the theory of reductive dual pairs due to Howe, and on the representation theory of Mickelsson algebras.
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Introduction to Sensitivity and Stability Analysis in Nonlinear Programming

1983-01-01
Anthony V. Fiacco

Loop Groups, Clusters, Dimers and Integrable Systems

2016-01-01
V. V. Fock , Andrey Marshakov
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Quantum flag varieties, equivariant quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi mathvariant="script">D</mml:mi></mml:math>-modules, and localization of quantum groups

2005-07-02
Erik Backelin , Kobi Kremnizer

A computation of universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$

2008-01-01
S. Khoroshkin , S. Pakuliak
We compute weight functions (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed … We compute weight functions (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in {EKP}.
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Cluster χ-varieties, amalgamation, and Poisson—Lie groups

2007-12-31
V. V. Fock , A. B. Goncharov
In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, … In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, as defined in []. In particular they are Poisson varieties. We define canonical Poisson maps of these varieties to the group G equipped with the standard Poisson—Lie structure defined by V. Drinfeld in [, ]. One of them maps to the group birationally and thus provides G with canonical rational coordinates.

Cluster algebras I: Foundations

2001-12-28
Sergey Fomin , Andrei Zelevinsky
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras. In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

Smoothness of certain metric projections on Hilbert space

1973-01-01
Richard B. Holmes
A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered … A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered within the context of more general Banach spaces.
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