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Irreducible representations of algebras Goodaire, Edgar George

Abstract

An element x of an associative algebra A is called diagonable provided A has a basis of characteristic vectors for the transformation ad x: a → ax - xa of A. This notion immediately generalizes to that of a diagonable subspace L of A. The centralizer A[sub O] of L plays an important role in the representation theory of A, for there is a one-to-one correspondence between the "λ-weighted" irreducible modules of A and of A[sub O]. In Chapters Two and Three, we first explore various ring-theoretic properties of A and A[sub O], and then use the results obtained to classify the diagonable elements in different algebras. We also give conditions under which all A-modules are weighted. The Cartan theory of Lie and Jordan algebras is linked in Chapter Four by the observation that Cartan subalgebras of simple finite dimensional Lie and Jordan algebras (over algebraically closed fields of characteristic 0) are diagonable subspaces of the respective universal enveloping algebras. Furthermore, in the Jordan case, the centralizer of a Cartan subalgebra is the centralizer of one of its elements and is a direct sum of complete matrix rings. Finally, we are able to show that the universal enveloping algebra of any simple Jordan algebra which contains an idempotent whose Peirce one-space is one-dimensional, is generated by its idempotents.

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