Lie algebras

Lie algebras

Definition

We have a nice axiomatic definition of a group and there is also a nice axiomatic definition of a Lie algebra; the Lie algebra of a matrix group furnishes us with example of things satisfying these axioms.

Definition:

A Lie algebra is a vector space (over a field; we'll usually use the real numbers or the complex numbers) little g equipped with an operation brackets going from little g times little g to little g satisfying:

  1. bilinearity (over our chosen field k), that is the ability to expand brackets: a X plus b Y brackets Z equals a X bracket Z plus b Y bracket Z, and X bracket (a Y + b Z) equals a X bracket Y plus b X bracket Z, for all X, Y and Z in little g and all a and b in the field k.

  2. for all X in little g we require X bracket X equals 0. This makes sense if you think about the commutator bracket, because X bracket X equals X X minus X X equals 0.

  3. the Jacobi identity: X bracket (Y bracket Z) plus Y bracket (X bracket X) plus Z bracket (X bracket Y) equals zero for all X, Y and Z in little g To remember this formula, note that the second and third terms are cyclic permutations of the first.

Remarks

Remark:

All three of these hold for the commutator bracket on matrices, but a Lie algebra doesn't need to have an underlying matrix product for which the bracket is given by commutator.

Remark:

  1. The property X bracket X equals zero implies that X bracket Y equals minus Y bracket X. To see this, note that X plus Y bracketed with itself equals X bracket X plus Y bracket Y plus X bracket Y plus Y bracket X and the three terms of the form something bracketed with itself all vanish.

  2. The converse also holds if we're working over a field of characteristic not equal to 2. In this case, X bracket X equals minus X bracket X implies 2 X bracket X equals 0, which implies X bracket X equals 0 as long as you can divide by 2. This implication fails if the field has characteristic 2 (e.g. for Z mod 2), so we assume the stronger axiom that X bracket X equals 0.

Lie subalgebras

Definition:

A Lie subalgebra little h inside little g is a subspace such that for all X and Y in little h, X bracket Y is also in little h.

All our examples are Lie subalgebras of little g l n R with commutator bracket. In particular, once you've proved the axioms for the commutator bracket, your proof shows that the axioms are satisfied for any Lie subalgebra (like the Lie algebra of a matrix group).

Ado's theorem

We will not prove the following (hard) theorem:

Theorem (Ado's theorem):

Any finite-dimensional Lie algebra over a field k is (isomorphic to) a Lie subalgebra of little g l n k for some n.

This shows that we really lose nothing by specialising to matrix groups at the level of their Lie algebras.

Pre-class exercise

Exercise:

Prove the Jacobi identity holds for the commutator bracket.