José I. Liberati

Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle

We classify positive energy representations with finite degeneracies of the Lie algebraW1+∞ and construct them in terms of representation theory of the Lie algebra\(\hat gl(\infty ,R_m )\) of infinites matrices with finite number of non-zero diagonals over the algebraR m =ℂ[t]/(tm+1). The unitary ones are classified as well. Similar results are obtained for the sin-algebras.

Finite growth representations of infinite Lie conformal algebras

We classify all finite growth representations of all infinite rank subalgebras of the Lie conformal algebra gc1 that contain a Virasoro subalgebra.

Matrix-valued bispectral operators and quasideterminants

We consider a matrix-valued version of the bispectral problem, that is, find differential operators and with matrix coefficients such that there exists a family of matrix-valued common eigenfunctions ψ(x, z): where f and Θ are matrix-valued functions. Using quasideterminants, we prove that the operators L obtained by non-degenerated rational matrix Darboux transformations from are bispectral operators, where and D is a diagonal matrix. We also give a procedure to find an explicit formula for the operator B extending previous results in the scalar case.

Representations of simple finite Lie conformal superalgebras of type W and S

We construct all finite irreducible modules over Lie conformal superalgebras of type W and S.

Classification of finite irreducible modules over the Lie conformal superalgebra CK 6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation superalgebra.

Irreducible modules over finite simple Lie conformal superalgebras of type K

We construct all finite irreducible modules over Lie conformal superalgebras of type K.We construct all finite irreducible modules over Lie conformal superalgebras of type K.

Quasifinite representations of classical Lie subalgebras of W∞,p

We show that there are exactly two anti-involutions σ± of the algebra of differential operators on the circle that are a multiple of p(t∂t) preserving the principal gradation (p∈C[x] non-constant). We classify the irreducible quasifinite highest weight representations of the central extension Dp± of the Lie subalgebra fixed by −σ±. The most important cases are the subalgebras Dx± of W∞ that are obtained when p(x) = x. In these cases, we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the Lie algebra of infinite matrices with finitely many nonzero diagonals over the algebra C[u]/(um+1) and its classical Lie subalgebras of C and D types.

The Central Extension Defining the Super Matrix Generalization of

We prove that the Lie superalgebra of regular differential operators on the superspace has an essentially unique non-trivial central extension.