✦ LIBER ✦

Representations of theq-Deformed Lie Algebra of the Group of Motions of the Euclidean Plane

✍ Scribed by Sergei D Silvestrov; Lyudmila B Turowska

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 467 KB
Volume: 160
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1998.3329

No coin nor oath required. For personal study only.

✦ Synopsis

Bounded an unbounded Hilbert space V -Representations of the q-deformed Lie algebra of the group of plan motions are studied for different choices of involutions. Integrable (well-behaved'') representations of the corresponding V -algebras are defined and described up to unitary equivalence. In the case of the V -algebras with quadratic involutions, analytically defined representations are introduced, and irreducible analytically defined representations are described up to unitary equivalence using dynamical systems. In the case of the V -algebras with involutions of the first order all analytically defined representations are shown to be onedimensional. For these V -algebras the problem of unitary classification of all representations defined on some dense invariant domain is shown to be equivalent to the unitary classification of arbitrary families of bounded self-adjoint operators. Integrable (well-behaved'') representations of these V -algebras are defined and described up to unitary equivalence.

1998 Academic Press

1. Introduction

In this paper we will study a family of complex unital associative algebras M q with generators z 1 , z 2 , z 3 and relations

q 1Â2 z 3 z 1 &q &1Â2 z 1 z 3 =0.

📜 SIMILAR VOLUMES

Computing Representations of a Lie Group

Computing Representations of a Lie Group via the Universal Enveloping Algebra

✍ Philip Feinsilver; René Schott 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 373 KB

We present an algorithm for the computation of representations of a Lie algebra acting on its universal enveloping algebra. This is a new algorithm which permits the effective computation of these representations and of the matrix elements of the corresponding Lie group. The approach is based on a m

The Support of an Irreducible Lie Algebr

The Support of an Irreducible Lie Algebra Representation

✍ Ivan Penkov; Vera Serganova 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 194 KB

We present an explicit description of the ᒅ-support supp M of any irreducible ᒅ-locally finite ᒄ-module M, where ᒄ is any finite-dimensional Lie algebra and ᒅ is an arbitrary nilpotent Lie subalgebra of ᒄ. If ᒅ contains a Cartan subalgebra of the semi-simple part of ᒄ, we reformulate the description

Generalized Restricted Lie Algebras and

Generalized Restricted Lie Algebras and Representations of the Zassenhaus Algebra

✍ Bin Shu 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 309 KB

Let F be a field of characteristic p ) 0, L a generalized restricted Lie algebra Ž . over F, and P L the primitive p-envelope of L. A close relation between Ž . L-representations and P L -representations is established. In particular, the irreducible -reduced modules of L for any g L\* coincide with

Versal Deformation of the Lie Algebra L2

✍ Alice Fialowski; Gerhard Post 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 120 KB

We investigate deformations of the infinite-dimensional vector-field Lie algebra spanned by the fields e s z iq 1 drdz, where i G 2. The goal is to describe the i base of a ''versal'' deformation; such a versal deformation induces all the other nonequivalent deformations and solves the deformation p

Irreducible Representations of the Lie-A

Irreducible Representations of the Lie-Algebra of the Diffeomorphisms of ad-Dimensional Torus

✍ S.Eswara Rao 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 234 KB

In this paper we investigate the irreducibility of certain modules for the Lie-algebra of diffeomorphisms of torus T d . The Lie-algebra of diffeomorw " 1 " 1 x Ž phisms can be described as the derivations of A s C t , . . . t see 1 d w x. w x RSS . We denote this Lie-algebra by Der A. Larsson L1 co

Explicit Constructions of the Fundamenta

Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras

✍ Robert G. Donnelly 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 237 KB

We give two constructions for each fundamental representation of sp 2 n, ‫ރ‬ . We also present quantum versions of these constructions. These are explicit in the sense of the Gelfand᎐Tsetlin constructions of the irreducible representations of Ž . Ž . gl n, ‫ރ‬ : we explicitly specify the matrix elem