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. / Science, Faculty of / Mathematics, Department of / Graduate

This dissertation explores numerical solutions for the cohomogeneity one Einstein and Ricci soliton equations when the principal orbits are SU(3)/T^2 and Sp(3)/Sp(1)^3. We present new numerical evidence for steady, expanding solitons as well as Einstein metrics with positive scalar curvature. In the case of steady solitons we produced a one-parameter family of solutions. In the expanding case, we generated a two-parameter family of solutions and in particular in the negative Einstein case we generated a one-parameter family of solutions. In the compact Einstein case we found numerical evidence for an in nite number of Einstein metrics. / Thesis / Master of Science (MSc)

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