Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of …
Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.
We construct a family of q -deformations of SU(2) for complex parameters q \neq 1 . For real q , the deformation coincides with the compact quantum SU _q (2) …
We construct a family of q -deformations of SU(2) for complex parameters q \neq 1 . For real q , the deformation coincides with the compact quantum SU _q (2) group. For q \notin \mathbb R , SU _q (2) is only a braided compact quantum group with respect to a twisted tensor product functor for C*-algebras with an action of the circle group.
For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions …
For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum groups, where the comultiplication takes values in a twisted tensor product. We show that compact braided C*-quantum groups yield compact quantum groups by a semidirect product construction.
We consider the category of C*-algebras equipped with actions of a locally compact quantum group. We show that this category admits a monoidal structure satisfying certain natural conditions if and …
We consider the category of C*-algebras equipped with actions of a locally compact quantum group. We show that this category admits a monoidal structure satisfying certain natural conditions if and only if the group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices. To prove this result we use only very natural properties imposed on considered monoidal structures. We assume that monoidal product is a crossed product, monoidal product of injective morphisms is injective and that monoidal product reduces to the minimal tensor product when one of the involved C*-algebras is equipped with a trivial action of the group. No a priori form of monoidal product is used.
Landstad-Vaes theory deals with the structure of the crossed product of a C$^*$-algebra by an action of locally compact (quantum) group. In particular it describes the position of original algebra …
Landstad-Vaes theory deals with the structure of the crossed product of a C$^*$-algebra by an action of locally compact (quantum) group. In particular it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on C$^*$-algebras. It is still possible to define crossed product (by weak action) and characterise the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.
We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted …
We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways. The first construction is based on certain pairs of representations of quantum groups which we call Heisenberg pairs because they generalise the Weyl form of the canonical commutation relations. The second construction uses covariant Hilbert space representations. We establish basic properties of the twisted tensor product and study some examples.
The main result of the Gelfand Naima~k theory of commutative C ~algebras can be stated in the following way. The category of commutative C'-algebras with unity and unital C'-homomorphisms is …
The main result of the Gelfand Naima~k theory of commutative C ~algebras can be stated in the following way. The category of commutative C'-algebras with unity and unital C'-homomorphisms is dual to the category of compact topological spaces and continuous maps. It is not difficult to extend this theory in order to include locally compact topological spaces. To this end one has to consider C'-algeb~as without unity. If A is a locally compact topological space, then C.(A ) denotes the C~-algebra of all complex continuous functions on A tending to 0 at infinity. In this case continuous maps from AI into A correspond to C'-homomorphisms from C~(A ) into C(A I ) satisfying certain condition (condition (C) of Section I) and the theory is comparable with that for compact spaces.
It is shown that all important features of a $\mathrm{C}^*$-algebraic quantum group $(A,\Delta)$ defined by a modular multiplicative $W$ depend only on the pair $(A,\Delta)$ rather than the multiplicative unitary …
It is shown that all important features of a $\mathrm{C}^*$-algebraic quantum group $(A,\Delta)$ defined by a modular multiplicative $W$ depend only on the pair $(A,\Delta)$ rather than the multiplicative unitary operator $W$. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary - without assuming existence of Haar weights.
We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S.L. Woronowicz. We prove that all the main results of …
We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S.L. Woronowicz. We prove that all the main results of the theory of manageable multiplicative unitaries remain true under this weaker condition. We also show that multiplicative unitaries arising naturally in the construction of some recent examples of non-compact quantum groups satisfy our condition, but fail to be manageable.
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a …
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a …
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.
'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q 2 ba, where q 2 = e -i ℏ is …
'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q 2 ba, where q 2 = e -i ℏ is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that [Formula: see text], where k = 0,1,2, ·. In this case, q is a root of 1. To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].
'az + b' is the group of affine transformations of complex plane [Formula: see text]. The coefficients a, [Formula: see text]. In quantum version a, b are normal operators such …
'az + b' is the group of affine transformations of complex plane [Formula: see text]. The coefficients a, [Formula: see text]. In quantum version a, b are normal operators such that ab = q 2 ba, where q is the deformation parameter. We shall assume that q is a root of unity, more precisely [Formula: see text], where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3, 18]. Then using the general theory we construct a C * -algebra A and a comultiplication Δ ∈ Mor (A, A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'az + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [18]. In the appendix, we briefly discuss the case of real q.
A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible …
A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL (2, C) consisting of the upper-triangular matrices. A description of the set of all 1-dimensional representations (the characters) of P is given. It turns out that the topological structure of this set is not the same as for the parabolic subgroup of the classical Lorentz group. The class of (in general non-unitary) representations of QLG induced by characters of its parabolic subgroup P is investigated. Representations act on spaces of smooth sections of (quantum) line boundles over the homogeneous space P\QLG (Gelfand spaces) as in the classical case. For any pair of Gelfand spaces the set of all non-zero invariant bilinear forms is described. This set is not empty only for certain pairs. We give a complete list of such pairs. Using this list we solve the problems of equivalence and irreducibility of the representation. We distinguish a class of Gelfand spaces carrying unitary representations of QLG.
A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that …
A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q 2 =e -iℏ . The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.
An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The …
An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The manageability is the property of multiplicative unitaries coming from the quantum group theory. For manageable multiplicative unitaries we reproduce all the essential results of the original paper of Baaj and Skandalis and much more. In particular the existence of the antipode and its polar decomposition is shown.
The main aim of this paper is to provide a proper mathematical framework for the theory of topological non-compact quantum groups, where we have to deal with non-unital C*-algebras. The …
The main aim of this paper is to provide a proper mathematical framework for the theory of topological non-compact quantum groups, where we have to deal with non-unital C*-algebras. The basic concepts and results related to the affiliation relation in the C*-algebra theory are recalled. In particular natural topologies on the set of affiliated elements and on the set of morphisms are considered. The notion of a C*-algebra generated by a finite sequence of unbounded elements is introduced and investigated. It is generalized to include continuous quantum families of generators. An essential part of the duality theory for C*-algebras is presented including complete proofs of many theorems announced in [17]. The results are used to develop a presentation method of introducing non-unital C*-algebras. Numerous examples related mainly to the quantum group theory are presented.
A new deformation of \mathit{SL}(2, \mathbf C) (considered as a real Lie group) is constructed and shown to have a Gauss type decomposition. The groups entering this decomposition are identified …
A new deformation of \mathit{SL}(2, \mathbf C) (considered as a real Lie group) is constructed and shown to have a Gauss type decomposition. The groups entering this decomposition are identified as E_μ(2) and its Pontryagin dual Ê_μ(2) . The whole group is the double group built over E_μ(2) .
For any number ν in the interval [-1, 1] , a C^* -algebra A , generated by two elements α and γ satisfying simple (depending on ν ) commutation relation, …
For any number ν in the interval [-1, 1] , a C^* -algebra A , generated by two elements α and γ satisfying simple (depending on ν ) commutation relation, is introduced and investigated. If ν=1 , then the algebra coincides with the algebra of all continuous functions on the group \textit{SU}(2) . Therefore, one can introduce many notions related to the fact that \textit{SU}(2) is a Lie group. In particular one can speak about convolution products, Haar measure, differential structure, cotangent boundle, left invariant differential forms. Lie brackets, infinitesimal shifts and Cartan Maurer formulae. One can also consider representations of \textit{SU}(2) . For ν< 1 , the algebra A is no longer commutative, however the notions listed above are meaningful. Therefore, A can be considered as the algebra of all “continuous functions” on a “pseudospace \textit{S}_\nu\textit{U}(2) ” and this pseudospace is endowed with a Lie group structure. The potential applications to the quantum physics are indicated.
An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The …
An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The manageability is the property of multiplicative unitaries coming from the quantum group theory. For manageable multiplicative unitaries we reproduce all the essential results of the original paper of Baaj and Skandalis and much more. In particular the existence of the antipode and its polar decomposition is shown.
The main aim of this paper is to provide a proper mathematical framework for the theory of topological non-compact quantum groups, where we have to deal with non-unital C*-algebras. The …
The main aim of this paper is to provide a proper mathematical framework for the theory of topological non-compact quantum groups, where we have to deal with non-unital C*-algebras. The basic concepts and results related to the affiliation relation in the C*-algebra theory are recalled. In particular natural topologies on the set of affiliated elements and on the set of morphisms are considered. The notion of a C*-algebra generated by a finite sequence of unbounded elements is introduced and investigated. It is generalized to include continuous quantum families of generators. An essential part of the duality theory for C*-algebras is presented including complete proofs of many theorems announced in [17]. The results are used to develop a presentation method of introducing non-unital C*-algebras. Numerous examples related mainly to the quantum group theory are presented.
In this paper we propose a simple definition of a locally compact quantum group in reduced form. By the word `reduced' we mean that we suppose the Haar weight to …
In this paper we propose a simple definition of a locally compact quantum group in reduced form. By the word `reduced' we mean that we suppose the Haar weight to be faithful. So in fact we define and study an arbitrary locally compact quantum group, represented on the L2-space of its Haar weight. For this locally compact quantum group we construct the antipode with polar decomposition. We construct the associated multiplicative unitary and prove that it is manageable in the sense of Woronowicz. We define the modular element and prove the uniqueness of the Haar weights. Following [15] we construct the reduced dual, which will again be a reduced locally compact quantum group. Finally we prove that the second dual is canonically isomorphic to the original reduced locally compact quantum group, extending the Pontryagin duality theorem. Dans cet article nous proposons une définition simple des groupes quantiques localement compacts et réduits. L'adjectif ‘réduit’ exprime l'hypothèse de fidélité du poids de Haar. Nous définissons et étudions des groupes quantiques localement compacts arbitraires représentés sur l'espace L2 du poids de Haar. Nous construisons l'antipode de ce groupe quantique localement compact, ainsi que sa décomposition polaire. Nous construisons l'unitaire multiplicatif associé et nous démontrons qu'il est maniable au sens de Woronowicz. Nous définissons l'élément modulaire et démontrons l'unicité des poids de Haar. En nous inspirant de [15] nous construisons le dual réduit, qui est de nouveau un groupe quantique localement compact et réduit. Finalement, nous démontrons que le groupe quantique bidual est isomorphe au groupe quantique de départ, ce qui constitue une généralisation du théorème de dualité de Pontryagin.
For any number ν in the interval [-1, 1] , a C^* -algebra A , generated by two elements α and γ satisfying simple (depending on ν ) commutation relation, …
For any number ν in the interval [-1, 1] , a C^* -algebra A , generated by two elements α and γ satisfying simple (depending on ν ) commutation relation, is introduced and investigated. If ν=1 , then the algebra coincides with the algebra of all continuous functions on the group \textit{SU}(2) . Therefore, one can introduce many notions related to the fact that \textit{SU}(2) is a Lie group. In particular one can speak about convolution products, Haar measure, differential structure, cotangent boundle, left invariant differential forms. Lie brackets, infinitesimal shifts and Cartan Maurer formulae. One can also consider representations of \textit{SU}(2) . For ν< 1 , the algebra A is no longer commutative, however the notions listed above are meaningful. Therefore, A can be considered as the algebra of all “continuous functions” on a “pseudospace \textit{S}_\nu\textit{U}(2) ” and this pseudospace is endowed with a Lie group structure. The potential applications to the quantum physics are indicated.
'az + b' is the group of affine transformations of complex plane [Formula: see text]. The coefficients a, [Formula: see text]. In quantum version a, b are normal operators such …
'az + b' is the group of affine transformations of complex plane [Formula: see text]. The coefficients a, [Formula: see text]. In quantum version a, b are normal operators such that ab = q 2 ba, where q is the deformation parameter. We shall assume that q is a root of unity, more precisely [Formula: see text], where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3, 18]. Then using the general theory we construct a C * -algebra A and a comultiplication Δ ∈ Mor (A, A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'az + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [18]. In the appendix, we briefly discuss the case of real q.
'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q 2 ba, where q 2 = e -i ℏ is …
'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q 2 ba, where q 2 = e -i ℏ is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that [Formula: see text], where k = 0,1,2, ·. In this case, q is a root of 1. To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].
The main result of the Gelfand Naima~k theory of commutative C ~algebras can be stated in the following way. The category of commutative C'-algebras with unity and unital C'-homomorphisms is …
The main result of the Gelfand Naima~k theory of commutative C ~algebras can be stated in the following way. The category of commutative C'-algebras with unity and unital C'-homomorphisms is dual to the category of compact topological spaces and continuous maps. It is not difficult to extend this theory in order to include locally compact topological spaces. To this end one has to consider C'-algeb~as without unity. If A is a locally compact topological space, then C.(A ) denotes the C~-algebra of all complex continuous functions on A tending to 0 at infinity. In this case continuous maps from AI into A correspond to C'-homomorphisms from C~(A ) into C(A I ) satisfying certain condition (condition (C) of Section I) and the theory is comparable with that for compact spaces.
A new deformation of \mathit{SL}(2, \mathbf C) (considered as a real Lie group) is constructed and shown to have a Gauss type decomposition. The groups entering this decomposition are identified …
A new deformation of \mathit{SL}(2, \mathbf C) (considered as a real Lie group) is constructed and shown to have a Gauss type decomposition. The groups entering this decomposition are identified as E_μ(2) and its Pontryagin dual Ê_μ(2) . The whole group is the double group built over E_μ(2) .
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a …
We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.
In this paper we associate to every reduced C * -algebraic quantum group (A, Δ) (as defined in [11]) a universal C * -algebraic quantum group (A u , Δ …
In this paper we associate to every reduced C * -algebraic quantum group (A, Δ) (as defined in [11]) a universal C * -algebraic quantum group (A u , Δ u ). We fine tune a proof of Kirchberg to show that every * -representation of a modified L 1 -space is generated by a unitary corepresentation. By taking the universal enveloping C * -algebra of a dense sub * -algebra of A we arrive at the C * -algebra A u . We show that this C * -algebra A u carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.
We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted …
We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways. The first construction is based on certain pairs of representations of quantum groups which we call Heisenberg pairs because they generalise the Weyl form of the canonical commutation relations. The second construction uses covariant Hilbert space representations. We establish basic properties of the twisted tensor product and study some examples.
We introduce some equivalent notions of homomorphisms between quantum groups that behave well with respect to duality of quantum groups. Our equivalent definitions are based on bicharacters, coactions, and universal …
We introduce some equivalent notions of homomorphisms between quantum groups that behave well with respect to duality of quantum groups. Our equivalent definitions are based on bicharacters, coactions, and universal quantum groups, respectively.
A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that …
A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q 2 =e -iℏ . The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.
This book deserves to become the standard work on group representations for theoretical physicists interested in elementary particles or special relativity. In keeping with the research needs of the times, …
This book deserves to become the standard work on group representations for theoretical physicists interested in elementary particles or special relativity. In keeping with the research needs of the times, the mathematical foundations of Lie algebras and of topological groups are thoroughly covered in the substantial mathematical chapters.
Let G be a locally compact group, $${\Gamma\subset G}$$ an abelian subgroup and let Ψ be a continuous 2-cocycle on the dual group $${\Hat\Gamma}$$ . Let B be a C*-algebra …
Let G be a locally compact group, $${\Gamma\subset G}$$ an abelian subgroup and let Ψ be a continuous 2-cocycle on the dual group $${\Hat\Gamma}$$ . Let B be a C*-algebra and $${\Delta_B\in{\rm Mor}\,(B,B\otimes{\rm C}_0(G))}$$ a continuous right coaction. Using Rieffel deformation, we can construct a quantum group $${({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}$$ and the deformed C*-algebra B Ψ. The aim of this paper is to present a construction of the continuous coaction $${\Delta_B^\Psi}$$ of the quantum group $${({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}$$ on B Ψ. The transition from the coaction Δ B to its deformed counterpart $${\Delta_B^\Psi}$$ is nontrivial in the sense that $${\Delta_B^\Psi}$$ contains complete information about Δ B . In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C*-algebraic quantum Minkowski space.
In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss …
In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several properties.
We define \ast -Hopf algebras \mathit{Fun}(\mathit{SL}_q(N, ℂ; ε_1,…, ε_N)) , \mathit{Fun}(O_q(N, ℂ; ε_1,…, ε_N)) and \mathit{Fun}(\mathit{Sp}_q(n, ℂ; ε_1,…, ε_{2n})) as the real complexifications of \ast -Hopf algebras \mathit{Fun}(\mathit{SU}_q(N, ℂ; ε_1,…, …
We define \ast -Hopf algebras \mathit{Fun}(\mathit{SL}_q(N, ℂ; ε_1,…, ε_N)) , \mathit{Fun}(O_q(N, ℂ; ε_1,…, ε_N)) and \mathit{Fun}(\mathit{Sp}_q(n, ℂ; ε_1,…, ε_{2n})) as the real complexifications of \ast -Hopf algebras \mathit{Fun}(\mathit{SU}_q(N, ℂ; ε_1,…, ε_N)) , \mathit{Fun}(O_q(N, ℂ; ε_1,…, ε_N)) and \mathit{Fun}(\mathit{Sp}_q(N, ℂ; ε_1,…, ε_{2n})) of [RTF] (for q > 0 ). Such construction can be done for each coquasitriangular (CQT) \ast -Hopf algebra \mathbf A . The obtained object \mathbf A^{ℂℝ} is also a CQT \ast -Hopf algebra. We describe the theory of corepresentations of \mathbf A^{ℂℝ} in terms of such a theory for \mathbf A .
The Haar measure on some locally compact quantum groups is constructed. The main example we treat is the az+b-group of Woronowicz. We also briefly consider some other examples (like the …
The Haar measure on some locally compact quantum groups is constructed. The main example we treat is the az+b-group of Woronowicz. We also briefly consider some other examples (like the ax+b-group). We get the first examples of a locally compact quantum group where the Haar measure is not invariant with respect to the scaling group.
Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there …
Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operator h, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each such h determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms.
The q-Poincaré group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B …
The q-Poincaré group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B is a braided-quantum group structure on the q-Minkowski space of four-momentum with braided-coproduct Δ_p=p⊗1+1⊗p. Here the necessary B is not a usual kind of quantum group, but one with braid statistics. Similar braided vectors and covectors V(R′), V*(R′) exist for a general R-matrix. The abstract structure of the q-Lorentz group is also studied.
A compact Hopf *-algebra is a compact quantum group in the sense of Koornwinder. There exists an injective functor from the category of compact Hopf *-algebras to the category of …
A compact Hopf *-algebra is a compact quantum group in the sense of Koornwinder. There exists an injective functor from the category of compact Hopf *-algebras to the category of compact Woronowicz algebras. A definition of the quantum enveloping algebra U q (sl(n,C)) is given. For quantum groups SU q (n) and SL q (n,C), the commutant of a canonical representation of the quantum enveloping algebra for q coincides with the commutant of the dual Woronowicz algebra for q -1 .