## Rui Philip Xiao, Department of Pure Mathematics, University of Waterloo

### “Universal enveloping algebras”

Any associative algebra can be made into a Lie algebra by introducing the commutator bracket [a,b] = ab-ba, a Lie algebra arising in this way is called a commutator algebra. To a Lie algebra L we can construct a unital associative algebra U(L), called the universal enveloping algebra of L, such that every Lie algebra homomorphism from L to some associative algebra factors through U(L) in a universal way. The Poincar-Birkhoff-Witt theorem gives a more concrete description of U(L) and shows that L can be viewed as a Lie subalgebra of the commutator algebra of U(L). In this talk I will define the universal enveloping algebra of a Lie algebra and show its existence and uniqueness, then prove the Poincar-Birkhoff-Witt theorem. If time permits I will talk about more consequences of the PBW theorem.