On some central operators for loop Lie superalgebras

Abstract

Let g be either a basic classical Lie superalgebra or gl(m,n) over the field of complex numbers C. For any associative, commutative, and finitely generated algebra A with unity, we consider the loop Lie superalgebra g⊗A. In [11], Rao defined a class of central operators for g⊗A and conjectured that these central operators, which generalizes the classical Gelfand invariants, generate the algebra U(g⊗A)g for g=gl(m,n). In this article we prove this conjecture and we also give a spanning set for U(g⊗A)G where g is the orthosymplectic Lie superalgebra and G is the corresponding orthosymplectic supergroup.