Symmetry and Integrability of Equations of Mathematical Physics − 2011

Dmytro Popovych (National Taras Shevchenko University of Kyiv, Ukraine)

Lie-orthogonal operators

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.