Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas

Abstract

Abstract. In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups.The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen's character sum formula for a quotient of two Verma modules (one of the main results of this article).Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).

Resolutions and characters of irreducible representations of the N = 2 superconformal algebra

“…There are two different types of Verma-like modules, called the massive and the topological (≡chiral) ones, of which the latter may appear as submodules of the former, but not vice versa; more precisely, it is the twisted 1 topological Verma modules that appear as submodules. The embedding structure [11] of massive N = 2 Verma modules is indeed more complicated than, e.g., in the well-known sℓ(2) Verma-module case [12,13]. 2 However, the admissible N = 2 representations that we mostly concentrate on in this paper are the quotients of massive Verma modules such that at least one charged singular vector [23] is necessarily quotiened away.…”

Differential-Operator Representations of S n and Singular Vectors in Verma Modules

“…Deodhar et al [2] generalized the results further to more general matrix Lie algebras. Rocha-Caridi and Wallach [19,20] generalized the results of Verma [22] and Bernstein et al [1] to a class of graded Lie algebras possessing a Cartan decomposition and obtained Jantzen's character formula corresponding to the quotient of two Verma modules. The resolutions of irreducible highest weight modules over rank-2 Kac-Moody algebras were constructed.…”

Past the Highest-Weight, and What You Can Find There

“…A twisted topological Verma module V h,t;θ is freely generated by L ≤−1 , H ≤−1 , G ≤θ−1 , and Q ≤−θ−1 from a twisted topological highest-weight vector subjected to annihilation conditions (15), where, in addition,…”

“…What is somewhat unusual about this correspondence, though, is the fact that on the sℓ(2) side such a 'cusp' state satisfies the same annihilation conditions as the highest-weight state of the module, whereas on the N = 2 side it satisfies twisted topological highest-weight conditions. Thus, to the well-known sℓ(2) singular vectors |MFF(r, s, k) ± , r, s ∈ N, given by the construction of 14 , there correspond the so-called topological singular vectors 18,19 |E(r, s, t) ± , t = k + 2, which satisfy twisted topological highest-weight conditions (15) with θ = ∓r respectively:…”

“…Before proceeding to more precise formulations, let us see whether the "essential equivalence" of the N = 2 and sℓ (2) representation theories comes as a news: -on the one hand, the two algebras appear to have little in common, since one is a rank-2 (bosonic) affine Lie algebra, while the other is a rank-3 super algebra that is not an affine Lie algebra; -on the other hand, the appearance of the two algebras in CFT is often 'correlated', they 'share' parafermionic theories, etc. * Talk presented at the NATO Advanced Research Workshop on Theoretical Physics 'New Developments in Quantum Field Theory', June [14][15][16][17][18][19][20]1997, Zakopane, and IV International Conference 'Conformal Field Theories and Integrable Models', Chernogolovka, June [23][24][25][26][27]1997. Surprisingly or not, establishing the equivalence requires introducing somewhat unusual sℓ (2) representations; more precisely, 1.…”