Comtrans algebras

Date
1991
Authors
Shen, Xiaorong
Major Professor
Advisor
Jonathan D. H. Smith
Committee Member
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Mathematics
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Mathematics
Abstract

A new class of algebras, the so-called comtrans algebras that have recently arisen from the solution of a problem in differential geometry (cf. (1)), is investigated. Simple comtrans algebras determined by Lie algebras and by pairs of matrices are characterized. The two classes separate, except for the vector triple product algebra. It is shown that the representation theory of a comtrans algebra E and the representation theory of an associative universal envoloping algebra \widetilde M[subscript] E of E are equivalent. It is also shown that the universal enveloping algebra of a comtrans algebra E over a field and the tensor algebra over (E \wedge E) \oplus (E \otimes E) \oplus (E \otimes E) are isomorphic;A comtrans algebra CT(E,[beta]) from a pair (E,[beta]) consisting of a unital module E over a commutative ring R with 1 and a bilinear form [beta]:E[superscript]2 → R is produced. A "transposed" comtrans algebra CT(E,[beta])[superscript][tau] is also given by the pair (E,[beta]). It is shown that, under mild restrictions on the underlying model E, the automorphism groups of the formed space (E,[beta]) and of the comtrans algebras CT(E,[beta]) and CT(E,[beta])[superscript][tau] coincide. Conditions for a comtrans algebra to be a "form algebra" are also given.

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