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Section 3.5 Components

Since a Lie algebra \(\gg\) is a vector space, we can expand any element \(X\in\gg\) in terms of a basis \(\{e_m\}\text{,}\) so

\begin{equation} X = X^m e_m\tag{3.5.1} \end{equation}

where we have introduced the Einstein summation convention that repeated indices are summed over unless otherwise stated. Thus, the commutator of two elements \(X,Y\in\gg\) can be expanded as

\begin{equation} [X,Y] = [X^m e_m,Y^n e_n] = X^m Y^n [e_m,e_n]\tag{3.5.2} \end{equation}

in terms of commutators of the basis elements. But each such commutator must lie in the Lie algebra, that is, we must have

\begin{equation} [e_m,e_n] = C^p{}_{mn} \,e_p\tag{3.5.3} \end{equation}

for some coefficients \(C^p{}_{mn}\text{,}\) known as structure constants for \(\gg\text{.}\) A Lie algebra is completely determined by its structure constants in any basis.

We digress briefly to discuss linear maps on vector spaces. Any such map on \(\gg\) has a matrix representation in a given basis. Explicitly, if \(A\) is a linear map on \(\gg\text{,}\) then we write

\begin{equation} A(e_m) = A^n{}_m e_n\tag{3.5.4} \end{equation}

so that

\begin{equation} A(X) = A(X^m e_m) = X^m A(e_m) = X^m A^n{}_m e_n\tag{3.5.5} \end{equation}

so that the column vector \(X^m\) of coefficients is mapped to the column vector \(A^n{}_m X^m\text{,}\) which is often written as \(AX\text{.}\) The matrix whose components are \(A^n{}_m\text{,}\) often written as \((A^n{}_m)\text{,}\) is the matrix of \(A\) in the given basis.

Since the action \(Y\longmapsto[X,Y]\) is a linear map for each \(X,Y\in\gg\text{,}\) we can express “commutators with \(X\)” as a matrix in terms of the given basis. In particular, the action of \(e_p\) on \(e_m\) is given by

\begin{equation} e_p(e_m) = [e_p,e_m] = C^n{}_{pm} \,e_n\tag{3.5.6} \end{equation}

so the matrix \(C_p\) representing “commutators with \(e_p\)” has components \(C^n{}_{pm}\) (with \(p\) fixed).

Remarkably, the matrices \(\{C_p\}\) also form a Lie algebra, namely the adjoint representation of \(\gg\text{,}\) as discussed further in Section 5.1. Here, we verify that these matrices satisfy the Jacobi identity (3.3.1).

The Jacobi identity on \(\gg\) in the form (3.3.4) implies that

\begin{equation} \bigl[[e_m,e_n],e_p\bigr] = \bigl[e_m,[e_n,e_p]\bigr] - \bigl[e_n,[e_m,e_p]\bigr] .\tag{3.5.7} \end{equation}

Writing out each term using (3.5.3) results in

\begin{equation} C^q{}_{mn} \,C^a{}_{qp} \,e_a = C^a{}_{mq} \,C^q{}_{np} \,e_a - C^a{}_{nq} \,C^q{}_{mp} \,e_a .\tag{3.5.8} \end{equation}

Since the coefficient of each basis element \(e_a\) must vanish independently, we can factor out \(e_a\) and rearrange terms to get

\begin{equation} C^a{}_{mq} \,C^q{}_{np} - C^a{}_{nq} \,C^q{}_{mp} = C^q{}_{mn} \,C^a{}_{qp}\tag{3.5.9} \end{equation}

or equivalently

\begin{equation} [C_m,C_n] = C_m C_n - C_n C_m = C^q{}_{mn} \,C_q .\tag{3.5.10} \end{equation}

We have shown that the matrices \(\{C_m\}\) have the same commutators (structure constants) as the basis elements \(\{e_m\}\text{!}\) Thus, the matrix algebra generated by \(\{C_m\}\) is isomorphic to the original Lie algebra \(\gg\text{!}\)

As the astute reader may have noticed already, this isomorphism is guaranteed by the form (3.3.4) of the Jacobi identity; all we have done is verify this isomorphism explicitly using components.