Proof of Proposition 3

Let g be a simple, finite-dimensional Lie superalgebra over C. These have been classified by V. Kac. Unless g is a Lie algebra or a Lie superalgebra of type osp(1, 2n), the category of finite-dimensional representations of g is not semisimple; q.v. [6]. This leads to a classification problem. For example, in [3], the representation theory of sl(m,n) is worked out by showing it is wild when m,n ≥ 2, and by giving an explicit description of the indecomposable finite-dimensional representations of sl(1, n). We show that the category of finite-dimensional representations of g is wild (i.e., as hard as classifying pairs of matrices) when g is of type W (0, n) with n ≥ 3. This is done by explicitly exhibiting enough extensions between simple modules. Secondly, we find the decomposition of the category of finite-dimensional representations into blocks. As an application, using an idea of Maria Gorelik, we prove that the centre of the universal enveloping algebra of g is trivial.