Homomorphisms of quantum groups (original) (raw)

Quantum families of quantum group homomorphisms

Communications in Algebra, 2016

The notion of a quantum family of maps has been introduced in the framework of C *-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.

Duality properties for quantum groups

Pacific Journal of Mathematics, 2011

Some duality properties for induced representations of enveloping algebras involve the character T rad g. We extend them to deformation Hopf algebras A h of a noetherian Hopf k-algebra A 0 satistying Ext i A0 (k, A 0) = {0} except for i = d where it is isomorphic to k. These duality properties involve the character of A h defined by right multiplication on the one dimensional free k[[h]]-module Ext d A h (k[[h]], A h). In the case of quantized enveloping algebras, this character lifts the character T rad g. We also prove Poincaré duality for such deformation Hopf algebras in the case where A 0 is of finite homological dimension. We explain the relation of our construction with quantum duality.

A Global Quantum Duality Principle for Subgroups and

2007

For a complex or real algebraic group G, with g := Lie(G) , quantizations of global type are suitable Hopf algebras F q [G] or U q (g) over C q, q −1. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G * and a dual Lie bialgebra g *. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called flat in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever. We end the paper with some explicit examples of application of our recipes.

A C*-ALGEBRAIC FRAMEWORK FOR QUANTUM GROUPS

International Journal of Mathematics, 2003

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.

The operator algebra approach to quantum groups

Proceedings of the National Academy of Sciences, 2000

A relatively simple definition of a locally compact quantum group in the C * -algebra setting will be explained as it was recently obtained by the authors. At the same time we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries and duality theory.

A global quantum duality principle for subgroups and homogeneous spaces

For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions. The global quantum duality principle (GQDP) - cf. [F. Gavarini, "The global quantum duality principle", J. Reine Angew. Math. 612 (2007), 17-33] - associates with any global quantization of G, or of g, a global quantization of g^*, or of G^*. In this paper we present a similar...

A global quantum duality principle for subgroups and homogeneous spaces, preprint

2012

Abstract. We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations: namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups are characterized as being the fixed points in this Galois ’ reciprocity. By the natural link between quantum subgroups and quantum homogeneous spaces then we argue a quantum duality principle for homogeneous spaces too, where quantum coisotropic spaces are the fixed elements in a Galois ’ reciprocity.

On Quantum Groups Co-Representations

We carry out a generalization of quantum group co-representations in order to encode in this structure those cases where non-commutativity between endomorphism matrix entries and quantum space coordinates happens.

The hidden group structure of quantum groups: Strong duality, rigidity and preferred deformations

Communications in Mathematical Physics, 1994

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyalproduct-like deformations are naturally found for all FRT-models on coefficients and C 00-functions. Strong rigidity (H^ = {0}) under deformations in the category of bialgebras is proved and consequences are deduced.

A quantum duality principle for subgroups and homogeneous spaces

Arxiv preprint math/0312289, 2003

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations: namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups are characterized as being the fixed points in this Galois' reciprocity. By the natural link between quantum subgroups and quantum homogeneous spaces then we argue a quantum duality principle for homogeneous spaces too, where quantum coisotropic spaces are the fixed elements in a Galois' reciprocity. As an application, we provide an explicit quantization of the homogeneous SL n (C) *-space of Stokes matrices, with the Poisson structure given by Dubrovin and Ugaglia.

Easy quantum groups

A closed subgroup G ⊂ u U + N is called easy when its associated Tannakian category C kl = Hom(u ⊗k , u ⊗l) appears from a category of partitions, C = span(D) with D = (D kl) ⊂ P , via the standard implementation of partitions as linear maps. The examples abound, and the main known subgroups G ⊂ U + N are either easy, or not far from being easy. We discuss here the basic theory, examples and known classification results for the easy quantum groups G ⊂ U + N , as well as various generalizations of the formalism, known as super-easiness theories, and the unification problem for them.

Introduction to quantum groups

This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group UNU_NUN has a free analogue UN+U_N^+UN+, whose standard coordinates uijinC(UN+)u_{ij}\in C(U_N^+)uijinC(UN+) are allowed to be free, and the closed subgroups GsubsetUN+G\subset U_N^+GsubsetUN+ can be thought of as being the compact quantum Lie groups. There are many interesting examples of such quantum groups, for the most designed in order to help with questions in quantum mechanics and statistical mechanics, and some general theory available as well, including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten integration. We discuss here the basic aspects of all this.

On the Structure of Inhomogeneous Quantum Groups

Communications in Mathematical Physics, 1997

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincaré groups [12]) we prove that our construction has correct 'size', find the R-matrices and the analogues of Minkowski space for G.

On a Morita equivalence between the duals of quantum and quantum

Advances in Mathematics, 2012

Let SU q Ô2Õ and Ö E q Ô2Õ be Woronowicz' q-deformations of respectively the compact Lie group SU Ô2Õ and the non-trivial double cover of the Lie group EÔ2Õ of Euclidian transformations of the plane. We prove that, in some sense, their duals are 'Morita equivalent locally compact quantum groups'. In more concrete terms, we prove that the von Neumann algebraic quantum groups L ÔSU q Ô2ÕÕ and L Ô Ö E q Ô2ÕÕ are unitary cocycle deformations of each other.

25 Years of Quantum Groups: from Definition to Classification

Acta Polytechnica, 2008

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

Duality for generalised differentials on quantum groups

Journal of Algebra

We study generalised differential structures Ω 1 , d on an algebra A, where A ⊗ A → Ω 1 given by a ⊗ b → adb need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs (Λ 1 , ω) where Λ 1 is a right module and ω a right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω : A + → Λ 1 in the category of right crossed (or Drinfeld-Radford-Yetter) modules over A. When A = U (g) the generalised left-covariant differential structures are classified by cocycles ω ∈ Z 1 (g, Λ 1). We then introduce and study the dual notion of a codifferential structure (Ω 1 , i) on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra (Ω, d) augmented by a codifferential i of degree −1. Here Ω is a graded super-Hopf algebra extending the Hopf algebra Ω 0 = A and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. We show how to construct such objects from first order data, with both a minimal construction using braided-antisymmetrizes and a maximal one using braided tensor algebras and with dual given via braidedshuffle algebras. The theory is applied to quantum groups with Ω 1 (Cq(G)) dually paired to Ω 1 (Uq(g)), and to finite groups in relation to (super) Hopf quivers.