Symmetry and Integrability of Equations of Mathematical Physics − 2011
Dmytro Popovych (National Taras Shevchenko University of Kyiv, Ukraine)
Basic properties of Lie-orthogonal operators on a finite-dimensional Lie algebra are discussed.
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 presented 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.