By William C. Waterhouse: pp. 164. DM.39.50; US$22Ā·20. (Springer-Verlag, Berlin, 1979.) AFFINE SETS AND AFFINE GROUPS (London Mathematical Society Lecture Note Series, 39) By D. G. Northcott: pp.285. Ā£9Ā·95. (Cambridge ā¦
By William C. Waterhouse: pp. 164. DM.39.50; US$22Ā·20. (Springer-Verlag, Berlin, 1979.) AFFINE SETS AND AFFINE GROUPS (London Mathematical Society Lecture Note Series, 39) By D. G. Northcott: pp.285. Ā£9Ā·95. (Cambridge University Press, Cambridge, 1980.)
We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of ā¦
We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories.
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Abstract We present an elementary proof of the fact that every torsor under an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness ā¦
Abstract We present an elementary proof of the fact that every torsor under an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral Tannakian categories.
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. ā¦
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. In each case, the result provides explicit inverse isomorphisms between two universally constructed bimonoids. We call these the Hoffman-Newman-Radford (HNR) isomorphisms. For a cocommutative comonoid, the free bimonoid on that comonoid is isomorphic to the free bimonoid on the same comonoid but with the trivial coproduct. The product is concatenation in both, but the coproducts differ, it is dequasishuffle in the former and deshuffle in the latter. An explicit isomorphism can be constructed in either direction, one direction involves a noncommutative zeta function, while the other direction involves a noncommutative Mƶbius function.These are the HNR isomorphisms. There is a dual result starting with a commutative monoid.In this case, the coproduct is deconcatenation in both, but the products differ, it is quasishuffle in the former and shuffle in the latter. Interestingly, these ideas can be used to prove that noncommutative zeta functions and noncommutative Mƶbius functions are inverse to each other in the lune-incidence algebra. There is a commutative analogue of the above results in which the universally constructed bimonoids are bicommutative. Now the HNR isomorphisms are constructed using the zeta function and Mƶbius function of the poset of flats. As an application, we explain how they can be used to diagonalize the mixed distributive law for bicommutative bimonoids. There is also a q-analogue, for q not a root of unity. In this case, the HNR isomorphisms involve the two-sided q-zeta and q-Mƶbius functions. As an application, we explain how they can be used to study the nondegeneracy of the mixed distributive law for q-bimonoids.
The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and ā¦
The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and field k, and conversely. Originally motivated by an attempt to find a first-order explanation for generic cohomology of algebraic groups, we study neutral tannakian categories as abstract first-order structures and, in particular, ultraproducts of them. One of the main theorems of this dissertation is that certain naturally definable subcategories of these ultraproducts are themselves neutral tannakian categories, hence tensorially equivalent to Comod_A for some Hopf algebra A over a field k. We are able to give a fairly tidy description of the representing Hopf algebras of these categories, and explicitly compute them in several examples. For the second half of this dissertation we turn our attention to the representation theories of certain unipotent algebraic groups, namely the additive group G_a and the Heisenberg group H_1. The results we obtain for these groups in characteristic zero are not at all new or surprising, but in positive characteristic they perhaps are. In both cases we obtain that, for a given dimension n, if p is large enough with respect to n, all n-dimensional modules for these groups in characteristic p are given by commuting products of representations, with the constituent factors resembling representations of the same group in characteristic zero. We later use these results to extrapolate some generic cohomology results for these particular unipotent groups.
An equivariant version of the twisted inverse pseudofunctor is defined, and equivariant versions of some important properties, including the Grothendieck duality of proper morphisms and flat base change are proved. ā¦
An equivariant version of the twisted inverse pseudofunctor is defined, and equivariant versions of some important properties, including the Grothendieck duality of proper morphisms and flat base change are proved. As an application, a generalized version of Watanabe's theorem on the Gorenstein property of the ring of invariants is proved.
This paper was motivated by a question of Vilonen, and the main results have been used by MirkoviÄ and Vilonen to give a geometric interpretation of the dual group (as ā¦
This paper was motivated by a question of Vilonen, and the main results have been used by MirkoviÄ and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a reductive group. We define a quasi-reductive group over a discrete valuation ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be an affine flat group scheme over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the identity component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the generic fiber is a reductive group, the special fiber is connected, and the group scheme is smooth over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in most cases, for example when the residue characteristic is not 2, or when the generic fiber and reduced special fiber are of the same type as reductive groups. We also obtain results about group schemes over a Dedekind scheme or a Noetherian scheme. We show that in residue characteristic 2 there are non-smooth quasi-reductive group schemes with generic fiber <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O Subscript 2 n plus 1"> <mml:semantics> <mml:msub> <mml:mi>SO</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {SO}_{2n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and they can be classified when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly Henselian.
In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras.Specifically, we construct a ā¦
In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras.Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids.The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors.One of them yields a contravariant adjunction together with the differentiation functor.Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids.By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids.Several examples and applications are presented along the exposition.
This paper gives methods to describe the adjoint orbits of G(or) on Lie(G)(or ) where or = o/p r (r ā N) is a finite quotient of the completion o ā¦
This paper gives methods to describe the adjoint orbits of G(or) on Lie(G)(or ) where or = o/p r (r ā N) is a finite quotient of the completion o of the ring of integers of a number field at a prime ideal p and G is a closed Z-subgroup scheme of GL n for an n ā N such that the Lie ring Lie(G)(o) is quadratic.The main result is a classification of the adjoint orbits in Lie(G)(o r+1 ) whose reduction mod p r contains a ā Lie(G)(or ) in terms of the reduction mod p of the stabilizer of a for the G(or )-adjoint action.As an application, this result is then used to compute the representation zeta function of the principal congruence subgroups of SL 3 (o).
We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which ā¦
We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton $\langle$On groups of linear recurrences, I$\rangle$ and by Aoki-Sakai $\langle$Mod $p$ equivalence classes of linear recurrence sequences of degree 2$\rangle$.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over <inline-formula content-type="math/mathml"> ā¦
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> form an abelian category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="HVec Subscript upper A"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HVec</mml:mtext> </mml:mrow> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\textrm {HVec}_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; the Fourier-Mukai transform yields an equivalence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="HVec Subscript upper A"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HVec</mml:mtext> </mml:mrow> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\textrm {HVec}_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="HVec Subscript upper A"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HVec</mml:mtext> </mml:mrow> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\textrm {HVec}_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the category of finite-dimensional representations of a commutative affine group scheme (the āaffine fundamental groupā of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.
Infinite comatrix corings Get access L. El Kaoutit, L. El Kaoutit Search for other works by this author on: Oxford Academic Google Scholar J. GĆ³mez-Torrecillas J. GĆ³mez-Torrecillas Search for other ā¦
Infinite comatrix corings Get access L. El Kaoutit, L. El Kaoutit Search for other works by this author on: Oxford Academic Google Scholar J. GĆ³mez-Torrecillas J. GĆ³mez-Torrecillas Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2004, Issue 39, 2004, Pages 2017ā2037, https://doi.org/10.1155/S1073792804134077 Published: 01 January 2004 Article history Received: 30 December 2003 Published: 01 January 2004 Revision received: 22 March 2004 Accepted: 19 May 2004
We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ā¦
We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The first class consists of those Hopf <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras that are connected graded as algebras, and the second class are those Hopf <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite GelāfandāKirillov dimension. It is shown that if the Hopf algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a connected graded Hopf algebra of finite GelāfandāKirillov dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a noetherian domain which is CohenāMacaulay, ArtināSchelter regular, and Auslander regular of global dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S squared equals normal upper I normal d Subscript upper H"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">S^2 = \mathrm {Id}_H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and it is CalabiāYau. Detailed information is also provided about the Hilbert series of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras of finite GelāfandāKirillov dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with connected coalgebra, the underlying coalgebra is shown to be ArtināSchelter regular of global dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of GelāfandāKirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis German g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(\mathfrak {g})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of ā¦
We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzatās freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is the first known case of the twenty year old conjecture.
We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able ā¦
We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor as many-sorted first order structures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show that the theory of a diagonalizable proalgebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is determined by the theory of the base field and the theory of the character group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Some initial steps towards a comprehensive study of types are also made.
This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal ā¦
This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal category. However, that is not the case here; the relevant concept is that of monads and algebras over monads. We construct a monad on the category of species, and observe that algebras over it are the same as monoids in species. Dually, we construct a comonad whose coalgebras are the same as comonoids in species. In addition, we construct a mixed distributive law between this monad and comonad such that bialgebras over the resulting bimonad are the same as bimonoids in species. Moreover, the mixed distributive law can be deformed by a parameter q such that the resulting bialgebras are the same as q-bimonoids. The above monad, comonad, bimonad have commutative counterparts which relate to commutative monoids, cocommutative comonoids, bicommutative bimonoids in species. We briefly discuss the Mesablishvili-Wisbauer rigidity theorem. As a consequence, the category of species is equivalent to the category of 0-bimonoids, as well as to the category of bicommutative bimonoids. These ideas are developed in more detail later. We extend the notion of species from a hyperplane arrangement to the more general setting of a left regular band (LRB).
We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky ā¦
We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne--Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas--Borne.
We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for ā¦
We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.
Abstract We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective ā¦
Abstract We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.
In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct ā¦
In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.
An attempt was made to make this a self-contained reading. The first three chapters are intended to provide the necessary background. Chapter one develops the tools needed from Galois Cohomology. ā¦
An attempt was made to make this a self-contained reading. The first three chapters are intended to provide the necessary background. Chapter one develops the tools needed from Galois Cohomology. Chapter two is a brief description of involutions, and in chapter three we define the notion of (linear) algebraic group, we give some examples and discuss some of their properties. In chapter four, we discuss some variants of the classical Skolem-Noether theorem, requiring only that the subalgebra have a unique faithful representation of full degree over a separable closure. We verify that we can extend every isomorphism to the whole algebra by means of inner automorphisms, just as in the classical case. Examples of algebras that satisfy this condition are simple algebras and commutative Frobenius algebras. In chapter five, we attach involutions to our algebras. We show that Skolem-Noether type results hold over a separable closure and we discuss some descent problems. Chapter six is a study of k-conjugacy classes of maximal k-tori, the main goal of this dissertation. We are able to give explicit descriptions of k-conjugacy classes in particular cases. This was done by applying the general formalism developed in the chapter.
We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of ā¦
We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group or, equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space.This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks.Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.
We will give an explicit description of the universal central extensions of Chevalley algebras over Laurent polynomial rings with n variables, which is a natural generalization of the result for ā¦
We will give an explicit description of the universal central extensions of Chevalley algebras over Laurent polynomial rings with n variables, which is a natural generalization of the result for n=l established in Garland [1], and which is obtained in a different way from Kassel [4].Using this, we will discuss about a certain class of GIM Lie algebras which are introduced by Slodowy [5] as a generalization of Kac-Moody Lie algebras.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$. In a recent paper, Bate, Martin, Rƶhrle and Tange show that every (smooth) subgroup of $G$ ā¦
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$. In a recent paper, Bate, Martin, Rƶhrle and Tange show that every (smooth) subgroup of $G$ is separable provided that the characteristic of $k$ is very good for $G$. Here separability of a subgroup means that its scheme-theoretic centralizer in $G$ is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of $G$. The aim of this paper is to prove this more general result. Moreover, we provide a condition on the characteristic of $k$ that is necessary and sufficient for the smoothness of all centralizers in $G$. We finally relate this condition to other standard hypotheses on connected reductive groups.
We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We ā¦
We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p $-adic fields which is $ H $-Galois for some Hopf algebra $ H $ then $ \OL $ is free over its associated order $ \AH $ in $ H $. If $ H $ is commutative, we show that this conclusion remains valid in ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. By combining these results we prove a generalisation of Noether's theorem to nonclassical Hopf-Galois structures on domestic extensions of number fields.
This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of ā¦
This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic $K$-theory.
In a previous article (Amano and Masuoka, 2005 Amano , K. , Masuoka , A. ( 2005 ). PicardāVessiot extensions of Artinian simple module algebras . J. Algebra 285 : ā¦
In a previous article (Amano and Masuoka, 2005 Amano , K. , Masuoka , A. ( 2005 ). PicardāVessiot extensions of Artinian simple module algebras . J. Algebra 285 : 743 ā 767 . [CSA] [Crossref], [Web of Science Ā®] , [Google Scholar]), the author and Masuoka developed a PicardāVessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS)D-module algebras, it generalizes and unifies the standard PicardāVessiot theories for linear differential and difference equations. The purpose of this article is to define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding PicardāVessiot group schemes.
This short survey article reviews our current state of understanding of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problemsare lis ted.
This short survey article reviews our current state of understanding of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problemsare lis ted.