Lie algebra of a matrix Lie group

For a Matrix Lie group $G$, its tangent space $\mathfrak{g}=T_{I}G$ is closed under the matrix commutator. Indeed, for $X,Y\in \mathfrak{g}$, the group commutatorlies in $G$, and the mixed derivative at $(0,0)$ of its matrix logarithm is $XY-YX$. Thus $[X,Y]=XY-YX$ belongs to $\mathfrak{g}$.

If the matrix logarithm identifies a neighborhood of the identity in a Matrix Lie group with a neighborhood of zero in its tangent subspace, the logarithm of the displayed group commutator belongs to that subspace. Divide it by $t^2$ and take the limit. Finite-dimensional linear subspaces are closed, proving closure under the matrix commutator without attempting to give a nonsmooth square-root reparametrization a velocity.

The endomorphisms of a vector space form a Lie algebra under the commutator $[A,B]=AB-BA$. It is the Lie algebra of the general linear group; the Jacobi identity follows by expanding associative products.

The real Lie algebra of the unitary group is $\mathfrak{u}(N)=\{X:X^{†}=-X\}$, of real dimension $N^2$. Differentiating unitarity gives necessity; exponentiating any skew-Hermitian matrix gives a curve in the group, proving sufficiency.

The real Lie algebra of the special unitary group consists of traceless skew-Hermitian matrices. Differentiating unitarity and determinant one gives the two constraints; exponentiating any such matrix proves their sufficiency. The commutator is again skew-Hermitian and traceless. Its real dimension is $n^2-1$, and its complexification is ${\mathfrak{sl}}_n(\mathbb{C})$.

With the defining two-dimensional matrix trace and inner product $⟨X,Y⟩=-\mathrm{tr}(XY)$, the displayed Pauli matrix basis is orthonormal. The Pauli matrix multiplication law gives the indicated structure constants of a Lie algebra. With the alternative form $-2\mathrm{tr}(XY)$, the orthonormal basis is $-i{\sigma }_a/2$ and the structure constant is one. The bracket and inner-product normalizations must be kept consistent in projected gauge-field formulas.

The compact real Lie algebra $\mathfrak{su}(3)$ consists of traceless anti-Hermitian $3\times 3$ matrices. Physicists often specify Hermitian generators $t_a$ instead, so the genuine Lie algebra elements are $i{t}_a$. The compact real algebra and its complexification ${\mathfrak{sl}}_3(\mathbb{C})$ should be distinguished.

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

// Target: lie-theory.bigb

A real basis of the unitary Lie algebra is $i{E}_{ii}$, $E_{ij}-E_{ji}$ and $i(E_{ij}+E_{ji})$ for $i<j$, where $E_{ij}$ are matrix units. These separately span the imaginary diagonal and real and imaginary upper off-diagonal entries.