20 credits at level HE6
This module is a sequel to the module 'Linear Algebra' at level 2. Building upon the work on vector spaces studied at level 2, the notion of an 'algebra' as a vector space with a product is introduced. Exterior algebras, Lie algebras and Clifford Algebras are constructed. Matrix groups and their tangent spaces are also studied.
1. Vector spaces with products
2. Exterior algebras of vector spaces
3. Lie algebras
4. Matrix groups and their tangent spaces
5. Normed real algebras
6. Tensor products and Clifford algebras
Approximately two-thirds of the available time will be devoted to lectures based on printed notes issued to students. The remainder will be devoted to supervised tutorial work.
There will be an in-class test of duration 90 minutes testing the first learning outcome in week 7 of the module. The remaining two learning outcomes will be assessed by an examination in week 14.
when you have successfully completed this module you will:
to demonstrate that you have achieved the learning outcome you will:
|1.||have an understanding of the notion of an 'algebra' as a vector space with a product||verify the algebra axioms for examples and derive results about specific examples of algebra|
|2.||have knowledge of the classication of matrix groups via their tangent spaces||determine the tangent spaces of various matrix groups and recognise the dimension of these|
|3.||have familiarity with a variety of algebras, including examples of Lie algebras, normed real algebras, and Clifford algebras||construct and derive results related to Lie algebras, normed real algebras, and Clifford algebras|
Your achievement of the learning outcomes for this module will be tested as follows:
Before taking this module you must have successfully completed the following:
No restrictions apply.
Andrew Baker, "Matrix Groups: An Introduction to Lie group Theory", Springer (2001)
Morton L Curtis, "Abstract Linear Algebra", Springer (1990)
Morton L Curtis, "Matrix Groups", 2nd edition, Springer (1984)
Karin Erdman & mark J. Wildon, "Introduction to Lie Algebras", Springer (2006)
Kristopher Tapp, "Matrix Groups for Undergraduates", AMS (2005)
Sergei Winitzki, "Linear Algebra via Exterior Products", lulu.com (2010)
Many of the books on the reading list for MAS2505 will also be useful.
|Host Subject Group:||Mathematics|
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