Abstract: We construct a type III factor by using the free product construction introduced by Voi... more Abstract: We construct a type III factor by using the free product construction introduced by Voiculescu and show that its core is $ L (F_ {\ infty})\ otimes B (H) $. We prove that $ M_2 (C)* L^\ infty\ lbrack 0, 1\ rbrack $ is a type $ III_\ lambda $ factor if M 2 (C) is endowed with a nontracial state (depending on $\ lambda $).
Locally Compact Quantum Groups and Groupoids, 2002
Continuing our research on extensions of locally compact quantum groups, we give a classification... more Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact quantum groups by the cocycle bicrossed product construction. We also clarify the notion of an extension of locally compact quantum groups by relating it to the concept of a closed normal quantum subgroup and the quotient construction. Finally, we describe the infinitesimal objects of locally compact quantum quantum groups with 2 and 3 generators -Hopf * -algebras and Lie bialgebras.
We study the homogeneous space of the quantum group Uq(n) related to the subgroup Uq(n m )( m<... more We study the homogeneous space of the quantum group Uq(n) related to the subgroup Uq(n m )( m<n ), classify its irreducible representations and get a formula for its in- variant integral. We also study the double cosets Uq(n m)nUq(n)=Uq(n m) and the hypergroup structure associ- ated with them.
We establish the equivalence of three approaches to the theory of finite dimensional quantum grou... more We establish the equivalence of three approaches to the theory of finite dimensional quantum groupoids. These are the generalized Kac algebras of T. Yamanouchi, the weak Kac algebras, i.e., the weak C * -Hopf algebras introduced by G. Böhm-F. Nill-K. Szlachányi which have an involutive antipode, and the Kac bimodules. The latter are an algebraic version of the Hopf bimodules of J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including founda... more We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.
We construct a one parameter deformation of the group of 2 × 2 upper triangular matrices with det... more We construct a one parameter deformation of the group of 2 × 2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C * -algebra and the dual comultiplication.
This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic anal... more This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic analysis on topological groups. These papers are known to he classical and form the basis of numerous contemporary researches into these fields.
We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algeb... more We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.
ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Pre... more ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
Journal of Physics A: Mathematical and General, 1998
The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets... more The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets U (1) \ SU q (2)/U (1)". They form a family (depending on a parameter q) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.
Abstract: We construct a type III factor by using the free product construction introduced by Voi... more Abstract: We construct a type III factor by using the free product construction introduced by Voiculescu and show that its core is $ L (F_ {\ infty})\ otimes B (H) $. We prove that $ M_2 (C)* L^\ infty\ lbrack 0, 1\ rbrack $ is a type $ III_\ lambda $ factor if M 2 (C) is endowed with a nontracial state (depending on $\ lambda $).
Locally Compact Quantum Groups and Groupoids, 2002
Continuing our research on extensions of locally compact quantum groups, we give a classification... more Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact quantum groups by the cocycle bicrossed product construction. We also clarify the notion of an extension of locally compact quantum groups by relating it to the concept of a closed normal quantum subgroup and the quotient construction. Finally, we describe the infinitesimal objects of locally compact quantum quantum groups with 2 and 3 generators -Hopf * -algebras and Lie bialgebras.
We study the homogeneous space of the quantum group Uq(n) related to the subgroup Uq(n m )( m<... more We study the homogeneous space of the quantum group Uq(n) related to the subgroup Uq(n m )( m<n ), classify its irreducible representations and get a formula for its in- variant integral. We also study the double cosets Uq(n m)nUq(n)=Uq(n m) and the hypergroup structure associ- ated with them.
We establish the equivalence of three approaches to the theory of finite dimensional quantum grou... more We establish the equivalence of three approaches to the theory of finite dimensional quantum groupoids. These are the generalized Kac algebras of T. Yamanouchi, the weak Kac algebras, i.e., the weak C * -Hopf algebras introduced by G. Böhm-F. Nill-K. Szlachányi which have an involutive antipode, and the Kac bimodules. The latter are an algebraic version of the Hopf bimodules of J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including founda... more We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.
We construct a one parameter deformation of the group of 2 × 2 upper triangular matrices with det... more We construct a one parameter deformation of the group of 2 × 2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C * -algebra and the dual comultiplication.
This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic anal... more This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic analysis on topological groups. These papers are known to he classical and form the basis of numerous contemporary researches into these fields.
We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algeb... more We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.
ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Pre... more ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
Journal of Physics A: Mathematical and General, 1998
The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets... more The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets U (1) \ SU q (2)/U (1)". They form a family (depending on a parameter q) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.
Uploads
Papers by L. Vainerman