Author: Eiko

Time: 2025-10-13 09:06:00 - 2025-10-13 09:06:00 (UTC)

Lecturer: Matthew Westaway

Note taker: Eiko

Topics to be covered:

  • Basic theory of Lie algebra, notations

  • Universal enveloping algebras, what they are and the PBW theorem, applications

  • Representation theory of complex simple Lie algebras

  • Central characters (related to representation theory), Harish-Chandra theorem

  • Category O

  • Lie algebras in characteristic \(p\)

  • Optional: Group schemes

Optional books (not following any of them strictly but using them as references):

Lie algebras, finite and affine type chap 9-12 by Carter

Introduction to Lie algebras by Erdmann-Wildon

Lie algebras and representation theory by Humphreys

When we get category O:

Representations of semisimple Lie algebras in the BGG category O by Humphreys

Lecture 1

Let \(\mathbb{K}\) be a field.

  • Definition A Lie algebra over \(\mathbb{K}\) consists of a \(\mathbb{K}\)-vector space \(\mathfrak{g}\) together with a bilinear map \([\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}\) such that

    • (skew-symmetry) \([x,x]=0\) for all \(x\in\mathfrak{g}\)

    • (Jacobi identity) \([x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\) for all \(x,y,z\in\mathfrak{g}\).

    A Lie algebra is called abelian if \([x,y]=0\) for all \(x,y\in\mathfrak{g}\). (This would mean the bracket is trivial, nothing interesting, essentially just a vector space.)

Ideals and subalgebras:

  • Definition A subalgebra of a Lie algebra \(\mathfrak{g}\) is a vector subspace \(\mathfrak{h}\subseteq\mathfrak{g}\) such that \([\mathfrak{h},\mathfrak{h}]\subseteq\mathfrak{h}\). An ideal of a Lie algebra \(\mathfrak{g}\) is a vector subspace \(\mathfrak{h}\subseteq\mathfrak{g}\) such that \([\mathfrak{h},\mathfrak{g}]\subseteq\mathfrak{h}\).

For a morphism of Lie algebras, its kernel is an ideal and image is a subalgebra.

For example, the kernel of the trace map \(\mathfrak{gl}_n(\mathbb{K})\to\mathbb{K}\) is an ideal, which is \(\mathfrak{sl}_n(\mathbb{K})\) (the special linear Lie algebra).

Adjoint Representation