Induced representations of the multiparameter Hopf Superalgebras U_uq(gl(m/n)) and U_uq(sl(m/n))

Abstract

We construct induced representations of the multiparameter Hopf superalgebras U uq (gl(m=n)) and U uq (sl(m=n)). The rst superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup GL(m=n). The second superalgebra is a Hopf subalgebra of the rst for a special choice of the parameters. The representations are labelled by m + n integer numbers, respectively m + n 1 complex numbers, and act in the space of formal power series of (m + n)(m + n 1)=2 noncommuting variables, of which mn are odd and the rest are even. These variables generate a q-deformation of a ag supermanifold of the supergroup GL(m=n), respectively SL(m=n).2

Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices

“…These quantum flag manifolds are studied intensively both from the algebraic and the operator algebraic viewpoint, see e.g. [88,87,24,25,21,26,23,37,54,13,77,16,81].…”

On the quantum superalgebra U_q(gl(m,n)) and its representations at roots of 1

“…Unlike in [25], where the quantum algebra is defined as a group functor, the quantum superalgebra in the present paper is defined as a quantum deformation of U(g). In the literature there are several such deformations ( [7,8,9,17,24,29]) that are different from the one used here. For instance, compared with the one in the present paper, the quantum deformation of U(g) in [2,29] has an additional generator.…”

“…For instance, compared with the one in the present paper, the quantum deformation of U(g) in [2,29] has an additional generator. Whereas the references [7] and [8] are concerned with two parameter deformations.…”

Remarks on Quantum Differential Operators

“…In fact, as pointed out by V. Dobrev, there is an important body of work involving very meaningful and general invariant q-difference operators connected to representations of Hopf algebras and intertwining; we mention in particular the impressive results on q-conformal invariant equations (see e.g. [32,33,34,35] and references there). In this paper we discuss only some simple examples and techniques for q-differential equations and the main point here is to examine (experimentally) some of the possible natural q-deformations of certain important classical differential equations (usually associated with integrable hierarchies).…”