Lie algebras

Definition

0.00 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:

0.30 A Lie algebra is a vector space (over a field; we'll usually use $\mathbf{R}$ or $\mathbf{C}$ ) $\mathfrak{g}$ equipped with an operation $[-,-]\colon\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ satisfying:

bilinearity (over our chosen field $k$ ), that is the ability to expand brackets:
$[aX+bY,Z]=a[X,Z]+b[Y,Z]\mbox{ and }[X,aY+bZ]=a[X,Y]+b[X,Z]\ \forall X,Y,Z\in% \mathfrak{g},\ a,b\in k$
for all $X\in\mathfrak{g}$ we require $[X,X]=0$ . This makes sense if you think about the commutator bracket, because $[X,X]=XX-XX=0$ .
2.40 the Jacobi identity:
$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0\ \forall X,Y,Z\in\mathfrak{g}.$ To remember this formula, note that the second and third terms are cyclic permutations of the first.

Remarks

Remark:

3.40 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:

4.20 The property $[X,X]=0$ implies that $[X,Y]=-[Y,X]$ . To see this, note that
$[X+Y,X+Y]=[X,X]+[Y,Y]+[X,Y]+[Y,X]$ and the three terms $[X+Y,X+Y],[X,X],[Y,Y]$ all vanish.
5.10 The converse also holds if we're working over a field of characteristic not equal to 2. In this case, $[X,X]=-[X,X]$ implies $2[X,X]=0$ , which implies $[X,X]=0$ as long as you can divide by 2. This implication fails if the field has characteristic 2 (e.g. for $\mathbf{Z}/2$ ), so we assume the stronger axiom that $[X,X]=0$ .

Lie subalgebras

Definition:

6.00 A Lie subalgebra $\mathfrak{h}\subset\mathfrak{g}$ is a subspace such that for all $X,Y\in\mathfrak{h}$ , $[X,Y]\in\mathfrak{h}$ .

6.40 All our examples are Lie subalgebras of $\mathfrak{gl}(n,\mathbf{R})$ 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):

7.24 Any finite-dimensional Lie algebra over a field $k$ is (isomorphic to) a Lie subalgebra of $\mathfrak{gl}(n,k)$ 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.