Irreducible *‐representations of Lie superalgebrasB(0,n) with finite‐degenerated vacuum

Abstract

The problem of getting irreducible *‐representations π of Lie superalgebras B(0,n), n=1,2, is studied, starting with a recently constructed family of linear representations in terms of differential operators on the space C∞N of CN ‐valued C∞ ‐functions. Equivalent formulation via creation‐annihilation operators of a para‐Bose system with n degrees of freedom is used, and the domain D of any π is shown to be a subset of C∞N containing a nonzero vacuum subspace. By assuming its dimension finite, the necessary conditions for existence of π are derived. The method is applied to the superalgebra B(0,1) and a one‐parameter family Π of nonequivalent irreducible *‐representations in terms of unbounded linear operators on L2(R+)⊗C2 is obtained. Each representation π∈Π has a nondegenerated vacuum and for all z∈B(0,1) satisfying z=z*, the operators π(z) are essentially self‐adjoint.