Quasi-finite Irreducible Graded Modules for the Virasoro-like Algebra

2013 ◽  
Vol 20 (02) ◽  
pp. 181-196 ◽  
Author(s):  
Weiqiang Lin ◽  
Yucai Su
Keyword(s):  
Weight Module ◽  
Highest Weight Module ◽  
Graded Modules ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Graded Module ◽  
Intermediate Series ◽  
Uniformly Bounded

In this paper, we consider the classification of irreducible Z- and Z2-graded modules with finite-dimensional homogeneous subspaces over the Virasoro-like algebra. We prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight quasi-finite irreducible modules. As a consequence, we determine all the modules with nonzero center. Finally, we prove that there does not exist any non-trivial Z-graded module of intermediate series.

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.

Quantum Groups: Singular Vectors and BGG Resolution

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 623-643 ◽  
Author(s):  
Fyodor Malikov
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Weight Module ◽  
Verma Modules ◽  
Highest Weight Module ◽  
Singular Vectors ◽  
Type Formula ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Dimensional Module

We prove existence of BGG resolution of an irreducible highest weight module over a quantum group, classify morphisms of Verma modules over a quantum group and find formulas for singular vectors in Verma modules. As an application we find cohomology of the quantum group of the type [Formula: see text] with coefficients in a finite-dimensional module.

Modules for loop Affine-Virasoro algebras

2020 ◽  
pp. 2150055 ◽  
Author(s):  
S. Eswara Rao
Keyword(s):  
Central Element ◽  
Weight Module ◽  
Verma Modules ◽  
Triangular Decomposition ◽  
Necessary And Sufficient Condition ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Integrable Module ◽  
Necessary And Sufficient

In this paper, we study the representations of loop Affine-Virasoro algebras. As they have canonical triangular decomposition, we define Verma modules and their irreducible quotients. We give necessary and sufficient condition for a irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either a highest weight module or a lowest weight module whenever the canonical central element acts non-trivially. At the end, we construct Affine central operators for each integer and they commute with the action of the Affine Lie 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.

Classification of Z-Graded Modules of the Intermediate Series over the q-Analog Virasoro-like Algebra

2010 ◽  
Vol 17 (02) ◽  
pp. 247-256
Author(s):  
Yina Wu ◽  
Weiqiang Lin
Keyword(s):  
Graded Modules ◽  
Direct Sum ◽  
Intermediate Series

In this paper, we complete the classification of Z-graded modules of the intermediate series over a q-analog Virasoro-like algebra L. We first construct four classes of irreducible Z-graded L-modules of the intermediate series. Then we prove that any Z-graded L-module of the intermediate series must be one of the modules constructed by us, or a direct sum of some trivial L-modules.

A Family of Non-weight Modules over the Virasoro Algebra

2020 ◽  
Vol 27 (04) ◽  
pp. 807-820
Author(s):  
Guobo Chen
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Necessary And Sufficient Conditions ◽  
Weight Module ◽  
Highest Weight Module ◽  
Isomorphism Classes ◽  
Highest Weight ◽  
Necessary And Sufficient ◽  
Weight Modules

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

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).

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