## The centralizer of a Cartan subalgebra of a Jordan algebra

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- by Edgar G. Goodaire PDF
- Trans. Amer. Math. Soc.
**235**(1978), 314-322 Request permission

## 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.

## References

- Edgar G. Goodaire,
*Irreducible representations of algebras*, Canadian J. Math.**26**(1974), 1118–1129. MR**349763**, DOI 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 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**

## Additional Information

- © Copyright 1978 American Mathematical Society
- 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