© 2001 by London Mathematical Society
© The London Mathematical Society
Plain Representations of Lie Algebras
Institute of Mathematics, National Academy of Sciences of Belarus Surganova 11, Minsk 220072, Belarus, baranov{at}im.bas-net.by
School of Mathematics, University of East Anglia Norwich NR4 7TJ, A.Zalesskii{at}uea.ac.uk
Received 15 December 1999.
In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.
We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and sl2-free (which means that L has no quotient isomorphic to sl2). The ground field F is always assumed to be algebraically closed and of characteristic 0.
Department of Mathematics and Computer Science, Leicester University, Leicester LE1 7RH