Around the center
Maria Gorelik, Vladimir Hinich, Vera Serganova
TL;DR
The paper addresses describing centers of enveloping algebras for Lie superalgebras by replacing finite group actions with Lagrangian equivalence relations on a space $V=\mathfrak{h}^*$, using a geometry-of-functions framework. It develops linear and Lagrangian relations, their composition, reduction, and discriminants, and introduces regularity notions that lead to a detectability result for semiregular relations. A key contribution is proving a general semiregularity-detectability theorem, enabling a case-free proof of Musson's results on centers by inductively analyzing strata and reductions. The approach, extended to weak generalized root systems (WGRS), yields a broad, structural method to understand centers and central characters in Lie superalgebras with potential applications beyond the specific cases studied.
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.