Cartan Subalgebras of glinfty

Let V be a vector space over a field K of characteristic zero and V* be a space of linear functionals on V which separate the points of V. We consider V (sleet)V* as a Lie algebra of finite rank operators on V, and set mathfrak{gl} (V,V*) := V \otimes V*. We define a Cartan subalgebra of \mathfrak{gl} (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 mathfrak{gl} (V,V*) under the assumption that K is algebraically closed. A subalgebra of mathfrak{gl} (V,V*) is a Cartan subalgebra if and only if it equals (bigoplus)j ( Vj (sleet)(Vj)* ) (oplus) (V^0 (otimes) V*^0) for some one-dimensional subspaces Vj (subseteq) V and (Vj)* subseteq V* with (Vi)* (Vj) = deltaij K and such that the spaces V*0 = (bigcap)j (Vj)\bot \subseteq V* and V^0 = (bigcap)j (bigl( (Vj)* (bigr)\bot (subseteq) V satisfy V*^0 (V^0) = {0}. We then discuss explicit constructions of subspaces Vj and (Vj)* as above. Our second main result claims that a Cartan subalgebra of mathfrak{gl} (V,V*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra mathfrak{h} which coincides with the maximal locally nilpotent mathfrak{h}-submodule of mathfrak{gl} (V,V*), and such that the adjoint representation of mathfrak{h} is locally finite.