Generalized Verma Modules over Lie Algebras of Weyl Type

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.

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 SINGULAR VECTORS OF THE TOPOLOGICAL CONFORMAL ALGEBRA

1996 ◽  
Vol 11 (25) ◽  
pp. 4597-4621 ◽  
Author(s):  
A. M. SEMIKHATOV ◽  
I. YU. TIPUNIN
Keyword(s):  
General Formula ◽  
Singular Vector ◽  
Conformal Algebra ◽  
General Construction ◽  
Verma Modules ◽  
Recursion Relations ◽  
Singular Vectors ◽  
Generalized Verma Modules ◽  
New Construction ◽  
A Chain

A general construction is found for “topological” singular vectors of the twisted N=2 superconformal algebra. It demonstrates many parallels with the known construction for affine sℓ(2) singular vectors due to Malikov–Feigin–Fuchs, but is formulated independently of the latter. The two constructions taken together provide an isomorphism between the topological and affine sℓ(2) singular vectors. The general formula for topological singular vectors can be reformulated as a chain of direct recursion relations that allow one to derive a given singular vector | S(r, s)〉 from the lower ones | S(r, s′<s)〉. We also introduce generalized Verma modules over the twisted N=2 algebra and show that they provide a natural setup for the new construction for topological singular vectors.

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