Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

2015 ◽  
Vol 10 (5) ◽  
pp. 1233-1242 ◽  
Author(s):  
Xiufu Zhang
Keyword(s):  
Virasoro Algebra ◽  
Weight Space ◽  
Finite Dimensional ◽  
Weight Modules

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 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 Supports of weight modules over Witt algebras

2011 ◽  
Vol 141 (1) ◽  
pp. 155-170 ◽  
Author(s):  
Volodymyr Mazorchuk ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebra ◽  
Weight Module ◽  
Root Space ◽  
Abelian Subalgebra ◽  
Space Decomposition ◽  
Finite Dimensional ◽  
Graded Lie Algebra ◽  
Alpha Beta ◽  
Weight Modules

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).

scholarly journals Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

2015 ◽  
Vol 15 (02) ◽  
pp. 1650029 ◽  
Author(s):  
Leandro Cagliero ◽  
Fernando Szechtman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight Space ◽  
Arbitrary Field ◽  
Nilpotent Radical ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
A Chain ◽  
Algebra Over A Field

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

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.

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