Alexander Sergeev

The invariant polynomials on simple Lie superalgebras

Chevalleys theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant polynomials on G with the algebra of W-invariant polynomals on H, where W is the Weyl group of G, (2) each G-invariant polynomial is a linear combination of the powers of traces tr r(x), where r is a finite dimensional representation of G. nNone of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalleys theorem so as to embrace Lie superalgebras. nChevalleys theorem for anti-invariant polynomials is also given.

The Howe duality and the projective representations of symmetric groups

The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S_n itself, coincide with the irreducible representations of a certain algebra A_n. Recently M.~Nazarov realized irreducible representations of A_n and Young symmetrizers by means of the Howe duality between the Lie superalgebra q(n) and the Hecke algebra H_n, the semidirect product of S_n with the Clifford algebra C_n on n indeterminates. nHere I construct one more analog of Young symmetrizers in H_n as well as the analogs of Specht modules for A_n and H_n.

Superanalogs of the Calogero Operators and Jack Polynomials

Abstract A depending on a complex parameter k superanalog of Calogero operator is constructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indeterminates and parameter k the operator constructed by Veselov, Chalykh and Feigin [2, 3]. For the operator SL is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs (GL(V )×GL(V ), GL(V )) and (GL(V ), OSp(V )), respectively. We will show that for the generic m and n the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of ; for k = 1, they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin’s integral on these superspaces.

Orthogonal polynomials of a discrete variable and Lie algebras of complex-size matrices

AbstractWe give a uniform interpretation of the classical continuous Chebyshev and Hahn orthogonal polynomials of a discrete variable in terms of the Feigin Lie algebran

Centralizer construction of the Yangian of the queer Lie superalgebra

mathfrak{g}mathfrak{l} (lambda )

Projective Schur Functions as Bispherical Functions on Certain Homogeneous Superspaces

n for λ∈ℂ. The Chebyshev and Hahn q-polynomials admit a similar interpretation, and orthogonal polynomials corresponding to Lie superalgebras can be introduced. We also describe quasi-finite modules overn

Enveloping Algebra of GL(3) and Orthogonal Polynomials

mathfrak{g}mathfrak{l} (lambda )