ABSTRACT
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 ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187919/82b7f1d0-8838-4dd2-9241-d60fd25bfe31/content/in9_u001.tif"/> ) where 𝔤 ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187919/82b7f1d0-8838-4dd2-9241-d60fd25bfe31/content/in9_u001.tif"/> = 𝔤0 ⊕ 𝔤1 but 𝔤0 is central in 𝔤 ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187919/82b7f1d0-8838-4dd2-9241-d60fd25bfe31/content/in9_u001.tif"/> . 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 ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187919/82b7f1d0-8838-4dd2-9241-d60fd25bfe31/content/in9_u001.tif"/> ) 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 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187919/82b7f1d0-8838-4dd2-9241-d60fd25bfe31/content/in9_u005.tif"/> 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.
