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UBC Theses and Dissertations

A general Cartan theory Foster, David Merriall

Abstract

Recent results of Jacobson and Barnes indicate that Lie, Jordan and alternative algebras may have a common Cartan theory. In this thesis, we show this is indeed the case. We also show that for certain classes of non-associative algebras, called E-classes, that possess an Engel function, a general Cartan theory is possible. In Chapter One, a generalization of nilpotence and solvability is introduced that permits our Cartan theory for E-classes. In Chapter Two, we construct Cartan subalgebras for alternative algebras based on a given Engel function. Jacobson's Cartan theory for Jordan algebras is given in Chapter Three along with our extensions of his results. We point out that the Engel function for alternative algebras and Jordan algebras coincides, and may be used to give the classical Cartan theory for Lie algebras Commutative power associative algebras are discussed in Chapter Four, and some results are obtained.

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