Quantum Mechanics for Mathematicians

Note that this linear mapad(X), which can be written as [X,·], can be thought
of as the infinitesimal version of the conjugation action

(·)→etX(·)e−tX

The Lie algebra homomorphism property ofadsays that

ad([X,Y]) =ad(X)◦ad(Y)−ad(Y)◦ad(X)

where these are linear maps ong, with◦composition of linear maps, so operating
onZ∈gwe have

ad([X,Y])(Z) = (ad(X)◦ad(Y))(Z)−(ad(Y)◦ad(X))(Z)

Using our expression foradas a commutator, we find

[[X,Y],Z] = [X,[Y,Z]]−[Y,[X,Z]]

This is called the Jacobi identity. It could have been more simply derived as
an identity about matrix multiplication, but here we see that it is true for a
more abstract reason, reflecting the existence of the adjoint representation. It
can be written in other forms, rearranging terms using antisymmetry of the
commutator, with one example the sum of cyclic permutations

[[X,Y],Z] + [[Z,X],Y] + [[Y,Z],X] = 0

Lie algebras can be defined much more abstractly as follows:

Definition(Abstract Lie algebra). An abstract Lie algebra over a fieldkis a
vector spaceAoverk, with a bilinear operation

[·,·] : (X,Y)∈A×A→[X,Y]∈A

satisfying

1. Antisymmetry:
   [X,Y] =−[Y,X]


2. Jacobi identity:

[[X,Y],Z] + [[Z,X],Y] + [[Y,Z],X] = 0

Such Lie algebras do not need to be defined as matrices, and their Lie bracket
operation does not need to be defined in terms of a matrix commutator (although
the same notation continues to be used). Later on we will encounter important
examples of Lie algebras that are defined in this more abstract way.