20191051 Quantum Mechanics II Tutorial 7
Teaching Assistant: Oz Davidi
December 23, 2018
1 Lie Algebra and Adjoint Representation
1.1 Structure Constants
Since the generators are closed under Lie bracket, we can define the structure constants fabc in the following way c [Xa,Xb] = ifabcX . (1.1)
Note that by definition fabc = −fbac. It is remarkable that fabc contains the whole information that we need in order to maintain the group multiplication law. It is important to emphasize that the actual value of fabc is basis dependent.
1.2 The Adjoint Representation
Let us first define a Lie algebra representation, and then look again at the adjoint representation which you already saw in class.
1 1.2 The Adjoint Representation 1 LIE ALGEBRA AND ADJOINT REPRESENTATION
Definition. A representation of a Lie algebra g is a linear map π : g −→ gl (n, C) that preserves the Lie product ∀X,Y ∈ g : π ([X,Y ]) = [π (X) , π (Y )] . (1.2)
A linear map between two Lie algebras that preserves the Lie product is called a Lie algebra homomorphism. If this map is invertible, it is called an isomorphism.
Definition. A Lie algebra representation π : g −→ gl (V ) is irreducible if there is no nontrivial subspace ∅= 6 W ⊂ V such that
∀X ∈ g : π (X) W ⊆ W. (1.3)
One of the most important representations is the adjoint. There are two equivalent ways to defined it.
1.2.1 Georgi (p. 48)
We plug Eq. (1.1) into the Jacobi identity, and find
0 = [[Xa,Xb] ,Xc] + [[Xc,Xa] ,Xb] + [[Xb,Xc] ,Xa] (1.4)
= ifabd [Xd,Xc] + ifcad [Xd,Xb] + ifbcd [Xd,Xa] (1.5)
= −fabdfdceXe − fcadfdbeXe − fbcdfdaeXe . (1.6)
This gives us the condition on the structure constants
fabdfdce + fcadfdbe + fbcdfdae = 0 . (1.7)
We define a set of matrices Ta such that
[Ta]bc ≡ −ifabc . (1.8)
2 1.2 The Adjoint Representation 1 LIE ALGEBRA AND ADJOINT REPRESENTATION
Then, by using the relation of the structure constants, Eq. (1.7), we get
[Ta,Tb]ce = [Ta]cd [Tb]de − [Tb]cd [Ta]de (1.9)
= −facdfbde + fbcdfade (1.10)
= −fcadfdbe − fbcdfdae (1.11)
= fabdfdce (1.12)
= ifabd [Td]ce . (1.13)
We see that
[Ta,Tb] = ifabcTc . (1.14)
Therefore, the structure constants themselves generate a representation. This representation is called the adjoint representation. We mentioned that the structure constant are basis dependent, which is obvious, remembering that the generators span a vector space. Let us see it explicitly. We define a scalar product on the linear space of the generators in the adjoint representation as Tr [TaTb]. This is a real symmetric matrix, and can hence be diagonalized with orthogonal matrix. By performing a linear transformation on the generators 0 Xa −→ Xa = SabXa , (1.15) with some orthogonal matrix S, Eq. (1.15) induces the following transformation of the structure constants 0 −1 fabc −→ fabc = SadSbefdegSgc . (1.16)
The matrices Ta of the adjoint representation transform by