Author:
BARANOV A. A.,ZALESSKII A. E.
Abstract
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
[sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2).
The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.
Cited by
5 articles.
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