Associated Varieties for Classical Lie Superalgebras
Let 𝔤 = 𝔤0 ⊕ 𝔤1 be a finite dimensional Lie superalgebra over an algebraically closed field K of characteristic zero. We consider a filtration on the enveloping algebra U(𝔤) such that the associated graded ring is isomoiphic to U( 𝔤 ˜ ) where 𝔤 ˜ = 𝔤0 ⊕ 𝔤1 but 𝔤0 is central in 𝔤 ˜ . This filtration was used by A. D. Bell to show that if a certain determinant d(𝔤) is nonzero, then U(𝔤) is prime. We show in this case that d(𝔤) defines the non‐Azumaya locus of U( 𝔤 ˜ ) provided dim 𝔤1 is even.
When 𝔤 is classical simple we study the associated graded ideal gr P of a primitive ideal P in U(𝔤) . We show that the radical g r ( P ) of gr P is prime. This is an analog of a result of Borho‐Brylinski and Joseph con ceming the irreducibility of the associated variety.