Primeness of the enveloping algebra of the special Lie superalgebras

Abstract.

A primeness criterion due to Bell is shown to apply to the universal enveloping algebra of the Cartan type Lie superalgebras S (V ) and $\widetilde {S}(V;t)$ when dim V is even. This together with other recent papers yields¶¶Theorem. Let L be a finite-dimensional simple Lie superalgebra over an algebraically closed field of characteristic zero. Then L satisfies Bell's criterion (so that U ( L) is prime and hence semiprimitive), unless L is of one of the types: b (n) for n≥ 3; W (n) for odd n≥ 5; S (n) for odd n≥ 3.¶¶ On the other hand, if dim V is odd then U (S(V )) is never semiprime.