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Special Algebraic Structures: Course Description

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This is a graduate course in algebra covering various specialized topics that have applications in geometry. There are no geometry prerequisites. We will only discuss structures at the level of vector spaces. Occasionally, we may give brief remarks about how these algebraic structures are defined on manifolds, and what the difficulties are, for the geometry/topology students, but there will be no geometry questions on the assignments. It is all just algebra.

*This course should be of interest to and could be taken by any graduate student in pure mathematics.*

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Prerequisites: Linear algebra, group theory, and ring theory. No geometry background will be assumed or needed.

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Main Topics:

tensor products, exterior algebra, and inner product spaces; complex structures; symplectic linear algebra; Lorentzian vector spaces; generalized complex and Dirac structures; Clifford algebras and the related Spin groups; normed real division algebras; cross products and calibrations; projective spaces; Jordan algebras. This covers most special algebraic structures that arise in geometry other than Lie algebras, which have a course to themselves.

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There are two main themes running through the second half of the course:

• Quaternions and octonions can be used to understand some specific structures in a concrete way, such as specific Spin groups, orthogonal groups, and Jordan algebras, in particular low dimensions.
• There is a way to generalize the Cauchy-Schwartz inequality from inner product spaces to spaces with calibrations using normed division algebras.

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