The centralizer of a Cartan subalgebra of a Jordan algebra

Author:
Edgar G. Goodaire

Journal:
Trans. Amer. Math. Soc. **235** (1978), 314-322

MSC:
Primary 16A64; Secondary 17C25, 17C10

DOI:
https://doi.org/10.1090/S0002-9947-1978-0460384-8

MathSciNet review:
0460384

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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.

- Edgar G. Goodaire,
*Irreducible representations of algebras*, Canadian J. Math.**26**(1974), 1118–1129. MR**349763**, DOI https://doi.org/10.4153/CJM-1974-104-0 - Edgar G. Goodaire,
*A classification of Jordan bimodules by weights*, Comm. Algebra**6**(1978), no. 9, 887–910. MR**470005**, DOI https://doi.org/10.1080/00927877808822273 - Nathan Jacobson,
*Structure and representations of Jordan algebras*, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR**0251099**

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Keywords:
Diagonable subspace,
weighted representation,
reduced Jordan algebra,
Cartan subalgebra,
universal enveloping algebra

Article copyright:
© Copyright 1978
American Mathematical Society