We study the representation theory of the superconformal algebra Wk(g, fθ) associated to a minimal gradation of g. Here, g is a simple finite-dimensional Lie superalgebra with a non-degenerate even supersymmetric invariant bilinear form. Thus, Wk(g, fθ) can be the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the BershadskyKnizhnik algebras, the N = 2 superconfor...

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions...

We study the nonstandard q-deformation U ′ q (so 4) of the universal enveloping algebra U (so 4) obtained by deforming the defining relations for skew-symmetric generators of U (so 4). This algebra is used in quantum gravity and algebraic topology. We construct a homomor-phism φ of U ′ q (so 4) to the certain nontrivial extension of the Drinfeld–Jimbo quantum algebra U q (sl 2) ⊗2 and show that...

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite-dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite-dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representa...

In this talk we introduce the notion of a generalized representation of a Jordan algebra with unit which has the following properties: 1) Usual representations and Jacobson representations correspond to special cases of generalized representations. 2) Every simple Jordan algebra has infinitely many nonequivalent generalized representations. 3) There is a one-to-one correspondence between irredu...

The symmetric group Sk possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of Sk itself, coincide with the irreducible representations of the algebra Ak generated by indeterminates τi,j for i 6= j, 1 ≤ i, j ≤ n subject to the relations τi,j = −τj,i, τ i,j = 1, τi,jτm,l = −τm,lτi,j if {i, j} ∩ {m, l} = ∅; τi,jτj,mτi,j = τj,m...

We discuss irreducible highest weight representations of the sl2 loop algebra and reducible indecomposable ones in association with the sl2 loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation ...

The decompositions of the Fock and Basic modules of the affine Lie algebra ŝlN into irreducible submodules of the Yangian algebra Y (glN ) are constructed. Each of the irreducible submodules admits the unique up to normalization eigenbasis of the maximal commutative subalgebra of the Yangian. The elements of this eigenbasis are identified with specializations of Macdonald symmetric functions wh...

We study a class of representations called “calibrated representations” of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type GLn. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra ĝl m . Moreover, we give a character formula of these irreducible modules in...

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard q-deformation U ′ q(son) (which does not coincide with the Drinfel’d-Jimbo quantum algebra Uq(son)) of the universal enveloping algebra U(son(C)) of the Lie algebra son(C) when q is not a root of unity. These representations are exhausted by irreducible representations...