Lie algebras

A Lie algebra is an algebra \(\mathfrak{g}\) over the field \(\mathbb{F}\), in which the bilinear operation is the Lie bracket. The Lie bracket \(\require{physics} \comm{\cdot}{\cdot}\) is a bilinear function that further obeys:

  1. alternativity \[\begin{equation} \forall T_a \in \mathfrak{g}: \comm{T_a}{T_a} = 0 \end{equation}\]

  2. the Jacobi identity \[\begin{equation} \forall T_a, T_b, T_c \in \mathfrak{g}: \comm{T_a}{\comm{T_b}{T_c}} + \comm{T_c}{\comm{T_a}{T_b}} + \comm{T_b}{\comm{T_c}{T_a}} = 0 \end{equation}\]

It can be shown that bilinearity and alternativity together imply anticommutativity: \[\begin{equation} \forall T_a, T_b \in \mathfrak{g}: \comm{T_b}{T_a} = - \comm{T_a}{T_b} \thinspace . \end{equation}\]

It is interesting to see that by defining the adjoint mapping \[\begin{equation} \text{ad}_{T_a}: \mathfrak{g} \to \mathfrak{g}: T_b \mapsto \text{ad}_{T_a} T_b = \comm{T_a}{T_b} \thinspace , \end{equation}\] the Jacobi identity can be written as \[\begin{equation} \text{ad}_{T_a} \comm{T_b}{T_c} = \comm{\text{ad}_{T_a} T_b}{T_c} + \comm{T_b}{\text{ad}_{T_a} T_c} \thinspace , \end{equation}\] which reminds us of how derivatives work.

In the physics community, the elements \(T_a, T_b, \cdots\) are called the generators of the algebra if they are a basis for the underlying vector space.

For the structure constants, the Jacobi identity implies \[\begin{equation} \sum_d^n (f_{bcd} f_{ade} + f_{abd} f_{cde} + f_{cad} f_{bde}) = 0 \thinspace . \end{equation}\]

It is interesting to note that every associative algebra \(A\) over a field \(\mathbb{F}\) admits a Lie algebra \(L(A)\) over the same field \(\mathbb{F}\) (both having the same underlying vector space \(V\)), by defining the Lie bracket as the commutator: \[\begin{equation} \comm{T_a}{T_b} = T_a T_b - T_b T_a \thinspace . \end{equation}\] The associative algebra \(A\) is then called the enveloping algebra of the Lie algebra \(L(A)\).

The most important examples of Lie algebras are those that are associated to a matrix Lie group. We have

  • \(\mathfrak{gl}(n, \mathbb{R})\): the Lie algebra of the \((n \times n)\)-matrices with real entries,
  • \(\mathfrak{sl}(n, \mathbb{R})\): the Lie algebra of the \((n \times n)\)-matrices with real entries and trace 0,
  • \(\mathfrak{o}(n) = \mathfrak{so}(n)\): the Lie algebra of the \((n \times n)\) skew-symmetric matrices with real entries,
  • \(\mathfrak{u}(n) = \mathfrak{su}(n)\): the Lie algebra of the \((n \times n)\) anti-Hermitian matrices. An important example is the Lie algebra \(\mathfrak{su}(2)\).

The relation between the matrix Lie groups \(G\) and their associated Lie algebras \(\mathfrak{g}\) is \[\begin{equation} \mathfrak{g} = \set{M \in \mathbb{F}^{n \times n}; \forall t \in \mathbb{R}: \exp(t M) \in G} \thinspace . \end{equation}\]