Deeper understanding of Lie theory requires representation theory. One thing is to get to know why it’s named adj(A)B for Lie Bracket?

In Lie theory, the Lie bracket is sometimes denoted as [A, B] and sometimes as ad(A)B or adj(A)(B). The notion of “adjoint” comes from the representation theory of Lie algebras.

The term “ad” is short for “adjoint”, which refers to the adjoint representation of a Lie algebra. The adjoint representation is defined by (ad x)(y) = [x, y].

Simply put, the “ad” or “adjoint” is used to denote the action of one element of the Lie algebra on another. The adjoint of an element (A) is a linear map from the Lie algebra to itself, which is given by taking the Lie bracket with (A).

Therefore, the adjoint of (A) acting on (B) is denoted as ad(A)B or adj(A)(B), which is equal to the Lie bracket [A, B].

The adjoint representation of a Lie group is a way the group acts on its corresponding Lie algebra. For any group element g in a Lie group G and any element B of its corresponding Lie algebra, the adjoint of g acting on B is defined as follows:

Ad_g(B) = gBg^(-1), which essentially is the conjugation of B by g.

The adjoint (Ad_g) of g is a map that takes any element B in g to its conjugation by g.” This shows the intimate connection between group conjugation and the adjoint action, which is fundamental in the study of the structure of Lie groups and Lie algebras.

The second point is it’s also a differential at identity.

The third point is that it’s also a Lie Bracket (cross product in essence).

Fourth, an insight by Mathemaniac, that Lie Bracket is also a “Directional Derivative“. Instead of derivative on vectors, it’s derivative on a “vector field” – a leap jump in understanding!