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.