Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras

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".

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Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

International Journal of Mathematics ◽

10.1142/s0129167x15500536 ◽

2015 ◽

Vol 26 (08) ◽

pp. 1550053 ◽

Cited By ~ 10

Author(s):

Christopher Sadowski

Keyword(s):

Lie Algebras ◽

Operator Algebras ◽

Intertwining Operators ◽

Natural Setting ◽

Affine Lie Algebras ◽

Universal Enveloping Algebras ◽

Enveloping Algebras ◽

Defining Relations ◽

Standard Formula ◽

Exact Sequences

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

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Generalized Verma Modules over Lie Algebras of Weyl Type

Algebra Colloquium ◽

10.1142/s1005386709000157 ◽

2009 ◽

Vol 16 (01) ◽

pp. 131-142

Author(s):

Bin Xin ◽

Yuezhu Wu

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Verma Module ◽

Total Order ◽

Verma Modules ◽

Additive Subgroup ◽

Highest Weight Module ◽

Highest Weight ◽

Generalized Verma Modules ◽

Proper Quotient

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500340 ◽

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.

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Equivalence of certain categories of modules for quantized affine lie algebras

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700002159 ◽

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.

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