The centralizer of a Cartan subalgebra of a Jordan algebra

Abstract:

If L is a diagonable subspace of an associative algebra A over a field $\Phi \;(L$ is spanned by commuting elements and the linear transformations ad $x:a \mapsto x - xa,x \in L$, are simultaneously diagonalizable), then a map $\lambda :L \to \Phi$ is said to be a weight of L on an A-module V if the space ${V_\lambda } = \{ v \in V:vx = \lambda (x)v\;{\text {for}}\;{\text {all}}\;x \in L\}$ is nonzero. It is shown that if A is finite dimensional semisimple and the characteristic of $\Phi$ is zero then the centralizer of L in A is the centralizer of an element $x \in A$ if and only if x distinguishes the weights of L on every irreducible A-module. This theorem can be used to show that for each representative V of an isomorphism class of irreducible A-modules and for each weight $\lambda$ of L on V, the centralizer of L contains the matrix ring ${D_{{n_\lambda }}},D = {\text {End}_A}V,{n_\lambda } = {\dim _D}{V_\lambda }$ and in fact is the direct sum of all such algebras. If J is a finite dimensional simple reduced Jordan algebra, one can determine precisely those x in J whose centralizer in the universal enveloping algebra of J coincides with the centralizer of a Cartan subalgebra. The simple components of such a centralizer can also be found and in fact are listed for the degree $J \geqslant 3$ case.