u(1) — compact unitary Lie algebra

The unitary Lie algebra $\mathfrak{u}(1)$ is the real vector space

\mathfrak{u}(1)=\{X\in M_{1}(\mathbb C):X^*+X=0\}.

with commutator bracket $[X,Y]=XY-YX$ . Here $X^*$ means conjugate transpose.

Dimension and scalar field

A skew-Hermitian diagonal has one real parameter per entry, while each off-diagonal pair has two. Consequently the real dimension is $1$ . It is the abelian line $i\mathbb R$ .

This object is a real Lie algebra. Multiplication by $i$ does not generally preserve the defining skew-adjoint condition; there is no inherited complex-linear category membership.