Primeness of the Enveloping Algebra of a Cartan Type Lie Superalgebra

We show that a primeness criterion for enveloping algebras of Le superalgebras discovered by Bell is applicable to the Cartan type Lie superal-gebras W(n), n even. Other algebras are considered but there are no deenitive answers in these cases. Allen Bell has shown in B] that if L is a nite-dimensional Lie superalgebra over a eld of characteristic zero, then the primeness of the universal enveloping algebra U(L) is implied by the nonsingularity of the product matrix ((f i ; f j ]). Here ff 1 ; : : :; f s g is a basis for the odd part L 1 of L, and the matrix is deened over the polynomial algebra S(L 0). Bell used this result to show that the universal enveloping algebra of a nite-dimensional classical simple Lie superalgebra is prime except possibly in the case of algebras of type b(n). This outstanding case was settled in the negative by a direct argument ((KK], Z]). An obvious next step is to consider the simple algebras of Cartan type. Our main (new) result here is that for even n 4, U(W(n)) is prime. A good basic reference for the properties of Cartan type Lie superalgebras is S].