# MA653 Lie Algebras

Term(s): Term 1

Commitment: 30 Lectures

Assessment: 100% Oral Exam

Formal registration prerequisites: None

Assumed knowledge: Linear algebra and ring theory from Year 2: MA251 Algebra I: Advanced Linear Algebra and MA249 Algebra II: Groups and Rings.

Useful background: Any third year algebra module will be useful to have more familiarity with complicated abstract algebra results and proofs. Some examples include: MA3E1 Groups and Representations, MA3G6 Commutative Algebra, MA377 Rings and Modules and MA3D5 Galois Theory.

Synergies: Any algebraic module will go well with Lie Algebras and the module MA4E0 Lie Groups will have some overlap but from a different, more analytic/topological perspective.

Content: Lie algebras are related to Lie groups, and both concepts have important applications to geometry and physics. The Lie algebras considered in this course will be finite dimensional vector spaces over $\C$ endowed with a multiplication which is almost never associative (that is, the products $(ab)c$ and $a(bc)$ are different in general). A typical example is the $n^2$ -dimensional vector space of all $n\times n$ complex matrices, with Lie product $[A,B]$defined as the commutator matrix $[A,B]=AB-BA$ . The main aim of the course is to classify the building blocks of such algebras, namely the simple Lie algebras of finite dimension over $\C$ .

Books:

J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1979

T.O. Hawkes, Lie Algebras, Notes available from Maths Dept.

N. Jacobson, Lie Algebras, Dover, 1979

K.Erdmann and M. Wildon, Introduction to Lie Algebras, Springer 2006