Lie-orthogonal operators

Details Lie-orthogonal operators Lie-orthogonal operators

2011-09-07

arxiv.org/abs/1109.1548v1

Mathematics

Rings and Algebras

17 pages

Scientific paper

Basic properties of Lie-orthogonal operators on a finite-dimensional Lie algebra are studied. In particular, the center, the radical and the components of the ascending central series prove to be invariant with respect to any Lie-orthogonal operator. Over an algebraically closed field of characteristic 0, only a solvable Lie algebra with solvability degree not greater than two possesses Lie-orthogonal operators whose all eigenvalues differ from 1 and -1. The main result of the paper is that Lie-orthogonal operators on a simple Lie algebra are exhausted by trivial ones. This allows us to give the complete description of Lie-orthogonal operators for semi-simple and reductive algebras, as well as a preliminary description of Lie-orthogonal operators on Lie algebras with nontrivial Levi-Mal'tsev decomposition. The sets of Lie-orthogonal operators of some classes of Lie algebras (the Heisenberg algebras, the almost Abelian algebras, etc.) are directly computed.

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