# Definition The special unitary group $SU(n)$ is the [[Identity Component]] of the [[Unitary Group|unitary group]] $U(n)$. More simply, it is the group of complex [[Unitary Operator|unitary matrices]] with [[Determinant|determinant]] $+1$. This [[Lie group]] has [[dimension]] $n^2-1$. # Lie Algebra The [[Lie Algebra]] of the [[Unitary Group]] $U(n)$, denoted $\mathfrak{u}(n)$ is the set of all $n\times n$ anti-Hermitian matrices. Enforcing the unit determinant condition, we get: $ \det{e^{tX}} = e^{t \text{tr}(X)} = 1 \implies \text{tr}(X) = 0 $ Thus, $\mathfrak{su}(n)$ is the set of $n \times n$ [[Trace|traceless]] [[Hermitian Operator|anti-Hermitian]] matrices. $\dim(\mathfrak{su}(n)) = n^2 - 1$, as one can easily verify. It is obviously the same dimension as the underlying [[Lie group]].