
Mathematics 2002
Indecomposable representations of Lie superalgebrasAbstract: In 1960's I. Gelfand posed a problem: describe indecomposable representations of any simple infinite dimensional Lie algebra of polynomial vector fields. Here, by applying the elementary technique of Gelfand and Ponomarev, a toy model of the problem is solved: finite dimensional indecomposable representations of vect(02), the Lie superalgebra of vector fields on the (02)dimensional superspace, are described. Since vect (02) is isomorphic to sl(12) and osp(22), their representations are also described. The result is generalized in two directions: for sl(1n) and osp(22n). Independently and differently J. Germoni described indecomposable representation of the series sl(1n) and several individual Lie superalgebras. Partial results for other simple Lie superalgebras without Cartan matrix are reviewed. In particular, it is only for vect(02) and sh(04) that the typical irreducible representations can not participate in indecomposable modules; for other simple Lie superalgebras without Cartan matrix (of series vect(0n), svect(0n)$, svect(0n)', spe(n) for n>2 and sh(0m) for m>4) one can construct indecomposable representations with arbitrary composition factors. Several tame open problems are listed, among them a description of odd parameters
