The [[Matrix Lie group]], $\mbox{SU}(n)$ is a subgroup of [[U(n)]] and the special group of [unitary matrices](Unitary%20operators.md#Unitary%20matrices) with [determinant](Determinants.md) 1. Thus the group elements are [complex matrix exponential](Complex%20matrix%20exponentials.md)s. # Lie algebras of $\mathrm{SU}(n)$ The corresponding [[Lie algebras]] of, $\mathfrak{su}(n)$, consists of all matrices with complex elements with the following constraints: 1. $X^{\dagger} = -X$ ([anti-Hermiticity](Anti-Hermitian%20operator.md)) 2. $\mathrm{tr}(X)=0$ constraint 1. follows directly from [properties](Matrix%20exponentials.md#Properties%20of%20Matrix%20Exponentials) [2.](Matrix%20exponentials.md#^799851) and [3.](Matrix%20exponentials.md#^d3c55b) of the matrix exponential since for the group elements to be [unitary](Unitary%20operators.md) it must mean that $(e^{X})^\dagger = e^{X^\dagger}=e^{-X}.$ Constraint 2. follows from the [matrix exponential diagonalization](Matrix%20exponentials.md#Diagonalizing%20Diagonalizable%2020matrices%20md%20matrix%20exponentials) where we find that [$\det{(e^{tX})}=e^{t\mbox{tr}(X)}.$](Matrix%20exponentials.md#^af041f) #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups