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Transactions of the American Mathematical Society

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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
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.


References [Enhancements On Off] (What's this?)

  • [1] Edgar G. Goodaire, Irreducible representations of algebras, Canad. J. Math. 26 (1974), 1118–1129. MR 0349763
  • [2] Edgar G. Goodaire, A classification of Jordan bimodules by weights, Comm. Algebra 6 (1978), no. 9, 887–910. MR 0470005
  • [3] 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0460384-8
Keywords: Diagonable subspace, weighted representation, reduced Jordan algebra, Cartan subalgebra, universal enveloping algebra
Article copyright: © Copyright 1978 American Mathematical Society