Cartan Subalgebras of

Abstract

AbstractLet V be a vector space over a field of characteristic zero and V* be a space of linear functionals on V which separate the points of V. We consider V ⊗ V* as a Lie algebra of finite rank operators on V, and set (V, V*) := V ⊗ V*. We define a Cartan subalgebra of (V, V*) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of (V;V*) under the assumption that is algebraically closed. A subalgebra of (V, V*) is a Cartan subalgebra if and only if it equals for some one-dimensional subspaces Vj ⊆ V and (Vj)* ⊆ V* with (Vi)* (Vj) = δij and such that the spaces . We then discuss explicit constructions of subspaces Vj and (Vj)* as above. Our second main result claims that a Cartan subalgebra of (V, V*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of (V, V*), and such that the adjoint representation of is locally finite.