
Symmetry and Integrability of Equations of Mathematical Physics − 2011
Dmytro Popovych (National Taras Shevchenko University of Kyiv, Ukraine)
Lieorthogonal operators
Abstract:
Basic properties of Lieorthogonal operators on a finitedimensional 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 Lieorthogonal operator. Over an algebraically closed field of characteristic 0,
only a solvable Lie algebra with solvability degree not greater than two possesses Lieorthogonal operators
whose all eigenvalues differ from 1 and 1. The main result presented is that Lieorthogonal operators on a
simple Lie algebra are exhausted by trivial ones. This allows us to give the complete description of
Lieorthogonal operators for semisimple and reductive algebras, as well as a preliminary description
of Lieorthogonal operators on Lie algebras with nontrivial LeviMal'tsev decomposition. The sets of
Lieorthogonal operators of some classes of Lie algebras (the Heisenberg algebras, the almost Abelian
algebras, etc.) are directly computed.

