Representations of the generalized Lie algebra

Journal of Mathematical Physics, 1995

Explicit expressions for the generators of the quantum superalgebra U q [gl(n/m)} acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a GePfand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of ^-number identities.

Journal of Mathematical Physics, 1994

Explicit expressions for the generators of the quantum superalgebra U q [gl(n/m)} acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a GePfand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of ^-number identities.

Communications in Mathematical Physics, 1996

We start with the observation that the quantum group SL q (2\ described in terms of the algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation, we develop a general method of constructing quantum groups with similar property. We also develop this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry out our method in detail for root systems of type SL(2)', as a byproduct, we find a new series of quantum groupsmetaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations ofSL q (2).

Journal of Physics A: Mathematical …, 1996

Communications in Algebra, 2021

We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras and denote by H q (f, g), related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras. In particular, our class now includes the enveloping algebra of the 3-dimensional Heisenberg Lie algebra and its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of quantum generalized Heisenberg algebras, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain some Lie-theoretic flavor. We work over a field of arbitrary characteristic, although our results on the representations require that it be algebraically closed.

Journal of Physics A: Mathematical and General, 2002

We show a local duality between the left and right finite-dimensional representations on the enveloping algebra of sl(2) q , the generalized Lie algebra introduced by Lyubashenko and Sudbery. This duality works in a general context of algebras satisfying certain algebraic properties; our goal in this paper is to prove that the enveloping algebra of sl(2) q satisfies these abstract properties.

Journal of Physics A: …, 1998

1995

It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's., This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantised enveloping algebra, and suggests a canonical Poincare-Birkhoff-Witt basis.

International Journal of Mathematics and Mathematical Sciences, 2005

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard q-deformation U q (so n ) (which does not coincide with the Drinfel'd-Jimbo quantum algebra U q (so n )) of the universal enveloping algebra U(so n (C)) of the Lie algebra so n (C) when q is not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations of U q (so n ) is proved.

Acta Applicandae Mathematicae, 1990

q-Analogs of the basic structures discussed in 'Lie Algebras and Recurrence Relations I' are presented. The q-Heisenberg-Weyl (qHW) and qsl(2) algebras are discussed in detail. Presently it is known that such structures are very closely tied in with the theory of quantum groups. Among other topics, coherent state representations and their interpretations as q-identities for q-Hermite and AI-Salam-Chihara (q-Meixner) polynomials are discussed. A discussion of Clebsch-Gordan coefficients for a qsu(2)type algebra is presented.