On the Block Structure of Supersolvable Restricted Lie Algebras

Abstract

Block theory is an important tool in the modular representation theory Ž w x. Ž w x. of finite groups cf. 18 . Apart from a few papers e.g. 17, 13, 16 dealing with restricted simple Lie algebras there apparently has been no effort to do the same for other classes of restricted Lie algebras despite a good Ž w x . knowledge of the simple modules cf. e.g. 21 for the solvable case . The aim of this paper is to develop the block theory for reduced universal enveloping algebras of a finite dimensional supersolvable restricted Lie algebra as far as possible in close analogy to modular group algebras. In order to organize the paper in a concise way, we include some open questions on tensor products of simple modules. Under certain conditions either on the Lie algebra or on the character they have affirmative answers which are decisively used in the proofs of our main results. But unfortunately this is not true in general as we show by an example. In the following, we are going to describe the contents of the paper in more detail. The second section provides the necessary background from the block theory of associative algebras which perhaps is not as well-known to Lie theorists. In particular, we stress that the block decomposition induces an Ž . Ž . equivalence relation ‘‘linkage relation’’ on the finite set of isomorphism classes of simple modules and indicate the proof of a cohomological characterization of the linkage relation which is fundamental for some of the following results. The latter can be used to give a combinatorial description of the linkage relation via the Gabriel quiver. We include some results on the number of blocks, resp. the principal block, that hold for any restricted Lie algebra.