Around the center

Abstract

The center of a semisimple Lie algebra can be described as the algebra of W-invariant functions on the dual of the Cartan subalgebra. The centers of many Lie superalgebras have a similar description, but the defining equivalence relation on the dual of the Cartan subalgebra is not given by a finite group action. Lagrangian equivalence relations that we introduce generalize the action of a subgroup of the orthogonal group. Using them, we present a new proof of a result by Ian Musson about the centers of Lie superalgebras. Our proof is not based on a case-by-case analysis.