Three lectures on quantum groups: representations, duality, real forms (original) (raw)

Elements of the theory of induced representations1 for quantum groups

We analyze the elements characterizing the theory of induced representations of Lie groups, in order to generalize it to quantum groups. We emphasize the geometric and algebraic aspects of the theory, because they are more suitable for generalization in the framework of Hopf algebras. As an example, we present the induced representations of a quantum deformation of the extended Galilei algebra in (1 + 1) dimensions.

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.

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.

On the representation categories of matrix quantum groups of type AAA

arXiv (Cornell University), 2005

A quantum groups of type A is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke symmetry.

Degenerate quantum general linear groups

Advances in Theoretical and Mathematical Physics, 2020

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by m+n-2 nonnegative integers and two arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit basis is constructed for each finite dimensional simple module. For all m and n, the degenerate quantum group has a natural irreducible representation acting on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of C(q)^(m+n) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from t...

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.

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.