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Plain Representations of Lie Algebras
Author(s) -
Baranov A. A.,
Zalesskiǐ A. E.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610701002101
Subject(s) - mathematics , lie algebra , associative property , pure mathematics , representation theory , fundamental representation , algebraically closed field , quotient , affine lie algebra , field (mathematics) , representation of a lie group , representation (politics) , non associative algebra , algebra over a field , lie conformal algebra , current algebra , politics , political science , law , weight
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 sl 2 ‐free (which means that L has no quotient isomorphic to sl 2 ). The ground field F is always assumed to be algebraically closed and of characteristic 0.