Part III (MMath/MASt)

• Basic Algebra: for Commutative Algebra, Algebraic Geometry, Algebraic Number Theory, Coxeter Groups
• Representation Theory : for Lie Algebras and their Representations, (Commutative Algebra, Coxeter Groups)
• Galois Theory: helpful (but not essential) for Commutative Algebra (small part of Coxeter Groups)
• Number Fields: helpful (but not essential) for Commutative Algebra
• Algebraic Geometry: helpful for Algebraic $D$-modules

## Basic Algebra

You will need this for the following Part III courses:

• Commutative Algebra
• Coxeter Groups: especially Modules
• Algebraic Geometry: especially polynomial rings
• Algebraic Number Theory

### Reality check

• A first course on Groups - you should be comfortable in working with abstract groups and examples. See e.g. Ia Groups. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.
• The definition of a vector space. See e.g. Ib Linear Algebra. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions. You will make use of many of the standard theorems from IB Linear Algebra in Finite dimensional Lie and associative algebras.

### Prerequisites

You need to be comfortable working with rings and modules, particularly rings of polynomials. For example, you should know:

• definitions and examples of: rings, subrings, ring homomorphisms, ideals, quotient rings, prime ideals, maximal ideals;
• factoring in rings: definitions and properties of zero-divisors, units, prime, and irreducible elements;
• Zorn's lemma and constructing prime ideals;
• examples and counterexamples relating fields, Euclidean domains, principal ideal domains, unique factorization domains, and integral domains; the statement of Gauss' lemma;
• definitions and examples of: modules, submodules, module maps, quotient modules, direct sums of modules, free modules;
• how to work with polynomial rings.

You can use the following exercises to check your present level and to guide your study. For the Example Sheet questions, we recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Useful books and resources

• Notes from catch-up workshop 2016, provided by Stacey Law, thank you!
• P.J. Cameron Introduction to Algebra, OUP.
• B. Hartley, T.O. Hawkes Rings, Modules and Linear Algebra: a further course in algebra, Chapman and Hall, 1970.
• "Rings and Things", a helpful summary of results by Gareth Taylor.
• Explanation relating some ring theory results to basic number theory results, by Gareth Taylor.

## Representation Theory

You will need this for the following Part III courses:

• (some parts for) Coxeter Groups
• (Commutative Algebra)
• Lie Algebras and their Representations
• Modular Representation Theory

### Reality check

Standard undergraduate level group theory (especially normal subgroups, conjugacy classes in the symmetric group). See e.g. Ia Groups. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

Linear algebra, see e.g. Ib Linear Algebra.

### Prerequisites

Basics of the representation theory of finite groups over the complex numbers. For example:

• Basic definitions and examples of representations, subrepresentations, irreducible representations, the group algebra and permutation representations.
• Schur's Lemma and some consequences. Maschke's theorem.
• Character Theory. More specifically, orthogonality and completeness of characters. The character table of a group.
• Lifting and Induction. Induced characters and Frobenius reciprocity.

For Coxeter Groups you will only need the first point on this list.

For Lie Algebras and their Representations and Coxeter Groups, some familiarity with tensor products, symmetric powers and exterior powers of vector spaces is also helpful e.g Chapter 10.4 and Chapter 11 (particulary 11.5) of Dummit and Foote (see below), or see Simon Wadsley's Part II Representation Theory notes (Section 5.2 for Tensors, Lecture 12 for general symmetric and exterior powers).

You can use the following exercises to check your present level and to guide your study.

• Exercises from the Representation Theory workshop 2015, provided by Nicolas Dupré (thank you!)
• Representation Theory Example Sheet 3 Questions 6, 7, 9 and Example Sheet 4 Questions 2, 3 for tensor products + symmetric and exterior powers

### Useful books and resources

• Lecture notes for the Part II course (as given in 2011), courtesy of Gareth Taylor.
• Simon Wadsley's online lecture notes for the Part II course (as given in 2012).
• Workshop notes the Representation Theory workshop 2015, provided by Nicolas Dupré (thank you!).
• G. D. James, M. W. Liebeck, Representations and characters of groups, Cambridge University Press, 1993.
• Dummit and Foote Abstract Algebra, 3rd edition, John Wiley and Sons, 2004 - Chpater 10.4 for tensors, Chapter 11 for symmetric and exterior powers.
• Lecture notes for the 2013 Part III course given by Stuart Martin, courtesy of Gareth Taylor (so you can see what level you might be expected to start at).
• A blog post by Tim Gowers about tensor products.
• Explanation of tensors and their universal properties How to conquer tensor-phobia by Jeremy Kun.

## Galois Theory

The Part III course Commutative Algebra will not assume any specific definitions or results from this area, but some exposure to this increases facility with the concepts of basic algebra, and provide a bit of "culture".

This section is on the Number Theory page.