The aim of this paper is to evaluate in terms of q-special functions the objects (intertwining map, fusion matrix, exchange matrix) related to the quantum dynamical Yang-Baxter equation (QDYBE) for infinite dimensional representations (Verma modules) of the quantized universal enveloping algebra Uq(g) in the case g = sl(2,C). This study is done in the framework of the exchange construction, ini...

Galilean W3 vertex operator algebra (VOA) GW3(cL,cM) is constructed as a universal enveloping of certain non-linear Lie conformal algebra. It proved that this simple by using the determinant formula vacuum module. The reducibility criterion for Verma modules given, and existence subsingular vectors demonstrated. Free field realization its highest weight are obtained within rank 4 lattice VOA.

Let K be a locally compact p-adic field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We investigate the category Cg of coadmissible modules over the p-adic Arens-Michael envelope Û(g) of U(g). Let p ⊆ g be a parabolic subalgebra. The main result gives a canonical equivalence between the classical parabolic BGG category of g relative to p and a certain expli...

We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac-Moody algebra ĝ in suitable categories of Harish-Chandra modules. We show that at the critical level these algebras are isomorphic to the algebras of differential forms on various spaces of opers associated to the Langlands dual Lie algebra of g, whereas away from the critical level they bec...

In the present paper we study representations of Jacobi algebra. More concretely, define, analogously to case semi-simple Lie algebras, Verma modules over algebra ${\cal G}_2$. We their reducibility and give explicit construction reducible exhibiting corresponding singular vectors. Using this information a complete classification modules. than exhibit interrelation embeddings between these Thes...

For every involution w of the symmetric group Sn we establish, in terms ofa special canonical quotient of the dominant Verma module associated with w, an effective criterion, which allows us to verify whether the universal enveloping algebra U(sln) surjects onto the space of all ad-finite linear transformations of the simple highest weight module L(w). An easy sufficient condition derived from ...

Kazhdan-Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. In particular, the value Px,w(1) is the dimension of the intersection cohomology sheaf of the Schubert variety Xw at the T -fixed point indexed by x [KL2]. It is also the multiplicity of a certain irreducible module in a Verma module [BeB...

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac–Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of sl2 (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103–113]. In the simpler case of A1 the for...

We define a notion of ghost centre of a Lie superalgebra g = g0 ⊕ g1 which is a sum of invariants with respect to the usual adjoint action (centre) and invariants with respect to a twisted adjoint action (“anticentre”). We calculate the anticentre in the case when the top external degree of g1 is a trivial g0-module. We describe the Harish-Chandra projection of the ghost centre for basic classi...

The Misra-Miwa v-deformed Fock space is a representation of the quantized affine algebra Uv(ŝl`). It has a standard basis indexed by partitions and the non-zero matrix entries of the action of the Chevalley generators with respect to this basis are powers of v. Partitions also index the polynomial Weyl modules for Uq(glN ) as N tends to infinity. We explain how the powers of v which appear in t...