Representations of classical Lie subalgebras of quantum pseudodifferential operators

Abstract

We show that there is a family of anti-involutions σϵ,k (ϵ=±1 and k∊Z), up to conjugation, of the Lie algebra Sq of quantum pseudodifferential operators preserving the principal gradation. We classify the irreducible quasifinite highest weight modules over the Lie subalgebras of Sq, fixed by minus σϵ,k(keven), and we realize them in terms of irreducible highest weight representations of the Lie algebra of infinite matrices with finitely many nonzero diagonals and its classical Lie subalgebras of B, C, and D types.