scholarly journals Singular vectors corresponding to imaginary roots in verma modules over affine Lie algebras.

1990 ◽  
Vol 66 ◽  
pp. 73
Author(s):  
F. Malikow
Keyword(s):  
Lie Algebras ◽  
Verma Modules ◽  
Affine Lie Algebras ◽  
Singular Vectors ◽  
Imaginary Roots

Imaginary Verma Modules for Affine Lie Algebras

1994 ◽  
Vol 37 (2) ◽  
pp. 213-218 ◽  
Author(s):  
V. M. Futorny
Keyword(s):  
Lie Algebras ◽  
Borel Subalgebra ◽  
Verma Modules ◽  
Affine Lie Algebras ◽  
Infinite Dimensions ◽  
Irreducible Modules

AbstractWe study a class of irreducible modules for Affine Lie algebras which possess weight spaces of both finite and infinite dimensions. These modules appear as the quotients of "imaginary Verma modules" induced from the "imaginary Borel subalgebra".

The branching problem for generalized Verma modules, with application to the pair (so(7), LieG2)

2014 ◽  
Vol 13 (07) ◽  
pp. 1450034
Author(s):  
Todor Milev ◽  
Petr Somberg
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Verma Modules ◽  
Singular Vectors ◽  
Weakly Compatible ◽  
Parabolic Subalgebras ◽  
Generalized Verma Modules ◽  
Branching Problem ◽  
Character Formulas

We consider the branching problem for generalized Verma modules Mλ(𝔤, 𝔭) applied to couples of reductive Lie algebras [Formula: see text]. Our analysis of the problem is based on projecting character formulas to quantify the branching, and on the action of the center of [Formula: see text] to construct explicitly singular vectors realizing the [Formula: see text]-top level of the branching. We compute explicitly the top part of the branching for the pair [Formula: see text] for both strongly and weakly compatible with i( Lie G2) parabolic subalgebras and a large class of inducing representations.

scholarly journals Equivalence of certain categories of modules for quantized affine lie algebras

2000 ◽  
Vol 69 (2) ◽  
pp. 162-175
Author(s):  
Vyacheslav M. Futorny ◽  
Duncan J. Melville
Keyword(s):  
Quantum Group ◽  
Lie Algebras ◽  
Verma Module ◽  
Verma Modules ◽  
Affine Lie Algebras ◽  
Finite Part ◽  
Moody Algebra ◽  
Finite Dimensional ◽  
Equivalence Of Categories ◽  
Category Of Modules

AbstractWe show that a quantum Verma-type module for a quantum group associated to an affine Kac-Moody algebra is characterized by its subspace of finite-dimensional weight spaces. In order to do this we prove an explicit equivalence of categories between a certain category containing the quantum Verma modules and a category of modules for a subalgebra of the quantum group for which the finite part of the Verma module is itself a module.

scholarly journals A construction of some ideals in affine vertex algebras

2003 ◽  
Vol 2003 (15) ◽  
pp. 971-980 ◽  
Author(s):  
Dražen AdamoviĆ
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Affine Lie Algebras ◽  
Explicit Formulas ◽  
Singular Vectors ◽  
Weyl Algebras ◽  
New Family

We study ideals generated by singular vectors in vertex operator algebras associated with representations of affine Lie algebras of typesAandC. We find new explicit formulas for singular vectors in these vertex operator algebras at integer and half-integer levels. These formulas generalize the expressions for singular vectors from Adamović (1994). As a consequence, we obtain a new family of vertex operator algebras for which we identify the associated Zhu's algebras. A connection with the representation theory of Weyl algebras is also discussed.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

A class of non-standard modules for affine Lie algebras

1987 ◽  
Vol 196 (3) ◽  
pp. 303-313 ◽  
Author(s):  
Nolan R. Wallach
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras
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