This module will be in the spirit of Algebra-I rather than Algebra-II. In fact, it could have even been called Very Advanced Linear Algebra. It will focus on explicit calculations with various linear algebraic objects, such as multilinear forms, which are a generalised version of linear functionals and bilinear forms. It could be useful in a range of modules.
Quaternions were discovered by Hamilton in 1843. We will introduce quaternions and develop computational techniques for 3D and 4D orthogonal transformations.
The word tensor was introduced by Hamilton at the time of discovery of quaternions. It used to mean the quaternionic absolute value. It acquired its modern meaning only in 1898, by which time Ricci had developed his Theory of Curvature (a prime example of tensor in Geometry). Later tensors spread not only to Algebra and Topology but also to some faraway disciplines such as Continuum Mechanics (elasticity tensor) and General Relativity (stress-energy tensor). Our study of tensors will concentrate on understanding the concepts and computation: we will not have time to develop any substantial applications.
When Elie Cartan discovered spinors in 1913, he could hardly imagine the role they would play in Quantum Physics. In 1928 Dirac wrote his celebrated electron equation, and since then there was no way back for spinors. According to Atiyah, “No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.”
As with tensors, our study of spinors will concentrate on understanding the concepts and computation: we will not have time to do any Physics. We plan to finish the module with Bott Periodicity for Clifford algebras.
This course will give the student a solid grounding in tensor algebra which is used in a wide range of disciplines.
There will be lecture notes. Some great books that the module will follow locally are:
Rotations, Quaternions, and Double Groups, by Simon L Altmann
The Algebraic Theory of Spinors, by Claude Chevalley
The Construction and Study of Certain Important Algebras, by Claude Chevalley
Rethinking Quaternions, by Ron Goldman
Quick Introduction to Tensor Analysis, by Ruslan Shapirov
Tensor Spaces and Exterior Algebras, by Takeo Yokonuma