Central Extensions and Derivations of Generalized Schrödinger-Virasoro Algebras

2012 ◽  
Vol 19 (04) ◽  
pp. 735-744 ◽  
Author(s):  
Wei Wang ◽  
Junbo Li ◽  
Bin Xin
Keyword(s):  
Lie Algebras ◽  
Natural Generalization ◽  
Additive Subgroup ◽  
Central Extensions ◽  
Infinite Dimensional

Let 𝔽 be a field of characteristic 0, G an additive subgroup of 𝔽, s ∈ 𝔽 such that s ∉ G and 2s ∈ G. A class of infinite-dimensional Lie algebras [Formula: see text] called generalized Schrödinger-Virasoro algebras was defined by Tan and Zhang, which is a natural generalization of Schrödinger-Virasoro algebras. In this paper, central extensions and derivations of [Formula: see text] are determined.

Verma Modules over Quantum Torus Lie Algebras

2010 ◽  
Vol 62 (2) ◽  
pp. 382-399 ◽  
Author(s):  
Rencai Lü ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Functions ◽  
Verma Modules ◽  
Central Extensions ◽  
Quantum Torus ◽  
Infinite Dimensional ◽  
Quantum Tori ◽  
Necessary And Sufficient

AbstractRepresentations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

scholarly journals Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

2009 ◽  
Vol 16 (04) ◽  
pp. 549-566 ◽  
Author(s):  
Shoulan Gao ◽  
Cuipo Jiang ◽  
Yufeng Pei
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Statistical Physics ◽  
Two Dimensional ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

INFINITE-DIMENSIONAL ℤN-INDEXED LIE ALGEBRAS AND THEIR SUPERSYMMETRIC GENERALIZATIONS

1989 ◽  
Vol 04 (16) ◽  
pp. 4295-4302 ◽  
Author(s):  
EDUARDO RAMOS ◽  
ROBERT E. SHROCK
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Explicit Representation ◽  
Mathematical Basis ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Different Types ◽  
Structure Constants ◽  
Infinite Dimensional Lie Algebra

We exhibit a new infinite-dimensional Lie algebra involving operators Ln with indices n ∈ ℤN and depending on a vector of structure constants, v. Two different types of central extensions are also presented. By constructing an explicit representation, we show that this algebra has a natural mathematical basis in terms of the algebra of infinitesimal diffeomorphisms of (S1)N. For N ≥ 2, the space of states is shown to have properties very different from those of the N = 1 (Virasoro) case. Supersymmetric generalizations are also given.

scholarly journals Descent constructions for central extensions of infinite dimensional Lie algebras

2006 ◽  
Vol 122 (2) ◽  
pp. 137-148 ◽  
Author(s):  
Arturo Pianzola ◽  
Daniel Prelat ◽  
Jie Sun
Keyword(s):  
Lie Algebras ◽  
Central Extensions ◽  
Infinite Dimensional

scholarly journals On complete reducibility for infinite-dimensional Lie algebras

2011 ◽  
Vol 226 (2) ◽  
pp. 1911-1972 ◽  
Author(s):  
Maria Gorelik ◽  
Victor Kac
Keyword(s):  
Lie Algebras ◽  
Infinite Dimensional ◽  
Complete Reducibility

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

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