📂Representation Theory

Structure Constants of Lie Algebras

Definition¹

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. Let $\left\{ X_{1}, \dots, X_{N} \right\}$ be a basis of $\mathfrak{g}$. The unique constants $c_{jk\ell}$ determined as below are called the structure constants of $\mathfrak{g}$.

$$ [X_{j}, X_{k}] = \sum_{\ell=1}^{N} c_{jk\ell}X_{\ell} $$

Properties

The following hold for all $j,k,\ell,m$.

$$ c_{jk\ell} + c_{kj\ell} = 0 \tag{1} $$

$$ \sum_{n} \left( c_{jkn}c_{n\ell m} + c_{k\ell n}c_{njm} + c_{\ell jn}c_{nkm} \right) = 0 \tag{2} $$

Explanation

$(1)$ holds because of the antisymmetry of the bracket $\left[ \cdot, \cdot \right]$, and $(2)$ holds by the Jacobi identity.

Since the bracket is bilinear, the bracket of any two elements $X = \sum_{j} a_{j} X_{j}$, $Y = \sum_{k} b_{k} X_{k}$ is entirely determined by the values of the brackets among the basis elements.

$$ [X, Y] = \sum_{j, k} a_{j} b_{k} [X_{j}, X_{k}] = \sum_{\ell} \left( \sum_{j, k} a_{j} b_{k} c_{jk\ell} \right) X_{\ell} $$

That is, once a basis is fixed, the structure constants completely determine the bracket of the Lie algebra, and therefore the entire structure of the Lie algebra.

The bracket of a Lie algebra is an abstract bilinear operation, but as seen above it is completely determined by the values among the basis elements alone. Since the structure constants are precisely a collection of those values, one can say that they compress the entire bracket operation over infinitely many pairs of elements into finitely many ($N^{3}$) numbers. This gives rise to the following advantages.

Computation

The bracket of any two elements reduces to arithmetic operations using the structure constants. Without having to handle the abstract operation directly, one can compute the Lie algebra using numerical data alone.

For example, the [basis $\left\{ X, H, Y \right\}$ of the special linear Lie algebra $\mathfrak{sl}(2; \mathbb{C})$ satisfies the commutation relations $[X, Y] = H$, $[H, X] = 2X$, $[H, Y] = -2Y$]. Using these, let us compute the bracket $[P, Q]$ of two elements $P = 2X + H$, $Q = X - Y$ in two ways and compare.

First, there is the method of viewing $P, Q$ as actual matrices and computing the commutator $PQ - QP$ directly by matrix multiplication.

$$ P = \begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}, \quad Q = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad [P, Q] = PQ - QP = \begin{pmatrix} -2 & 2 \\ 2 & 2 \end{pmatrix} = 2X - 2H + 2Y $$

On the other hand, one can forget what $P, Q$ actually are and expand using bilinearity (the distributive law) with only the coordinates and the commutation relations above, that is, the structure constants alone.

$$ [2X + H, X - Y] = 2\underbrace{[X, X]}_{0} - 2\underbrace{[X, Y]}_{H} + \underbrace{[H, X]}_{2X} - \underbrace{[H, Y]}_{-2Y} = 2X - 2H + 2Y $$

The results of the two methods are the same, but the second method did not multiply matrices even once. As long as one has the bracket values on basis pairs (the structure constants) and arithmetic operations, one can compute any bracket without even knowing that the Lie algebra is realized as matrices.

Generalization

Structure constants are not a concept confined to Lie algebras. Since the product $\times$ of any algebra over a field $A$ is also bilinear, one can define structure constants $c_{ijk}$ as below with respect to a basis $\left\{ e_{1}, \dots, e_{n} \right\}$.

$$ e_{i} \times e_{j} = \sum_{k=1}^{n} c_{ijk} e_{k} $$

The structure constants of a Lie algebra are the special case where the product is taken to be the bracket. The structure constants of a general algebra have no constraints, but in a Lie algebra the additional conditions of properties $(1)$ and $(2)$ are imposed because of the antisymmetry of the bracket and the Jacobi identity.

Example

Let the basis $\left\{ X, H, Y \right\}$ of the special linear Lie algebra $\mathfrak{sl}(2; \mathbb{C})$ be set as $(X_{1}, X_{2}, X_{3}) = (X, H, Y)$. From the commutation relations $[X, Y] = H$, $[H, X] = 2X$, $[H, Y] = -2Y$ computed below, the nonzero structure constants are as follows.

$$ c_{132} = 1, \quad c_{211} = 2, \quad c_{233} = -2 $$

The rest are determined by the antisymmetry of property $(1)$. For instance, $c_{312} = -1$, $c_{121} = -2$, $c_{323} = 2$.