Cartan subalgebras of gl∞

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 0 V. as a Lie algebra of finite rank operators on V, and set gl(V, V-*) := V circle times V-*. We define a Cartan subalgebra of 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 gl(V, V.) under the assumption that K is algebraically closed. A subalgebra of gl(V, V-*) is a Cartan subalgebra if and only if it equals circle plus (j)( V-j circle times (V-j)(*)) circle plus (V(0)circle times V-*(0)) for some one-dimensional subspaces V-j subset of or equal to V and (V-j)(*) subset of or equal to V-* with (V-i)(*) (V-j) = delta(ij)K and such that the spaces V-*(0) =boolean AND(j)(V-j)(perpendicular to) subset of or equal to V-* and V-0 = boolean AND(j) ( (V-j)(*))(perpendicular to) subset of or equal to 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 gl(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 gl(V, V-*), and such that the adjoint representation of h is locally finite.