scholarly journals CLASSIFICATION OF IRREDUCIBLE WEIGHT MODULES OVER THE TWISTED HEISENBERG–VIRASORO ALGEBRA

2010 ◽  
Vol 12 (02) ◽  
pp. 183-205 ◽  
Author(s):  
RENCAI LÜ ◽  
KAIMING ZHAO
Keyword(s):  
Virasoro Algebra ◽  
The Other ◽  
Simple Modules ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Highest Weight Modules ◽  
Intermediate Series ◽  
Weight Modules

In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose nonzero weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

scholarly journals Classification of irreducible integrable highest weight modules for current Kac–Moody algebras

2016 ◽  
Vol 16 (07) ◽  
pp. 1750123 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Direct Sums ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Highest Weight Modules ◽  
Weight Modules

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

scholarly journals Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

2012 ◽  
Vol 55 (3) ◽  
pp. 697-709 ◽  
Author(s):  
Xiangqian Guo ◽  
Rencai Lu ◽  
Kaiming Zhao
Keyword(s):  
Direct Summand ◽  
Virasoro Algebra ◽  
Additive Subgroup ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Intermediate Series ◽  
Complex Number Field ◽  
Weight Modules

AbstractLet G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

scholarly journals Characterization of Simple Highest Weight Modules

2013 ◽  
Vol 56 (3) ◽  
pp. 606-614 ◽  
Author(s):  
Volodymyr Mazorchuk ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebras ◽  
Positive Root ◽  
Virasoro Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Heisenberg Algebras ◽  
Higher Rank ◽  
Highest Weight Modules ◽  
Weight Modules

Abstract.We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.

scholarly journals UNITARY REPRESENTATIONS OF NONCOMPACT QUANTUM GROUPS AT ROOTS OF UNITY

2001 ◽  
Vol 13 (08) ◽  
pp. 1035-1054 ◽  
Author(s):  
H. STEINACKER
Keyword(s):  
Quantum Groups ◽  
Classical Case ◽  
Unitary Representations ◽  
Roots Of Unity ◽  
Classical Symmetry ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Frobenius Map ◽  
Highest Weight Modules ◽  
Weight Modules

Noncompact forms of the Drinfeld–Jimbo quantum groups [Formula: see text] with [Formula: see text], [Formula: see text] for si=±1 are studied at roots of unity. This covers [Formula: see text], su(n,p), so*(2l), sp(n,p), sp(l,ℝ), and exceptional cases. Finite dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius map is determined, and the meaning of the extra quasi-classical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q→1.

scholarly journals Representations of the Twisted Heisenberg–Virasoro Algebra at Level Zero

2003 ◽  
Vol 46 (4) ◽  
pp. 529-537 ◽  
Author(s):  
Yuly Billig
Keyword(s):  
Lie Algebra ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Highest Weight ◽  
Level Zero ◽  
Maximal Submodule ◽  
Highest Weight Modules ◽  
Weight Modules

AbstractWe describe the structure of the irreducible highest weight modules for the twisted Heisenberg–Virasoro Lie algebra at level zero. We prove that either a Verma module is irreducible or its maximal submodule is cyclic.

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