In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig’s a-function is formulated (and proved for type A). The second part deals with the extension algebra of Verma modules. I...

Modules over affine Lie superalgebras G are studied, in particular, for G = ̂ OSP (1, 2). It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most importance are the generalized Kac-Kazhdan formula and the Malikov-Feigin-Fuchs construction, which give the weights and the explicit form of the singular vectors in the Verma m...

Let g(G,λ) denote the deformed generalized Heisenberg-Virasoro algebra related to a complex parameter λ≠−1 and an additive subgroup G of C. For total order on that is compatible with addition, Verma module over defined. In this paper, we completely determine irreducibility these modules.

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on a integro-differential expression. This expression belongs to a class generated by means of Virasoro vector fields acting on the KdV field. The “almost” rational KdV fields are described in terms of a geometrical locus of complex points. A class of solutions of this locus has recently appea...

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of Coxeter groups and also by the geometry of Schubert cells in the flag manifold (see [8]). The purpose of this paper is to give the proof of their conjecture. The method employed here is to associate ...

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let gI be the corresponding semi-simple subalgebra of g. Denote by WI the Weyl group of gI and let w◦ and w I ◦ be the longest elements of W and WI , respectively. In this paper we show that the answer to Kostant’s problem, i.e. whether the universal enveloping algebra surjects on...

This text is an extended version of a part of my 1980 Trieste preprint [16]. In this preprint I gave, in particular, an expression for the action of a Lie superalgebra g on the g-module coinduced from an h-module V for the case where both g and V are finite-dimensional and g decomposes (as a linear space, not Lie superalgebra) into a direct sum g− ⊕ h with g− a Lie superalgebra. The importance ...

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When modules are two-dimensional, we categorical action blob algebra on derived categories these dg-algebras which intertwines $\mathfrak{sl}_2$. From above derive categorification algebra.