# (special) unitary Lie algebra

Let $V$ be a vector space over a field $K$ admitting an involution $\sigma:K\rightarrow K$, and let $B:V\times V\rightarrow{\mathbb{F}}$ be a http://planetmath.org/node/SesquilinearFormsOverGeneralFieldshermitian form relative to $\sigma$. Then the unitary Lie algebra ${\mathfrak{u}}(V,B)$, or just ${\mathfrak{u}}(V)$, consists of the linear transformations $T$ satisfying

 $B(Tx,y)+B(x,Ty)=0,$

for all $x,y\in V$. This is a Lie algebra over $k=\{\alpha\in K\mid\alpha^{\sigma}=\alpha\}$, but not over $K$ in the case that $K\neq k$ (because $B$ is linear in the first, but not in the second variable).

The special unitary Lie algebra ${\mathfrak{s}u}(V,B)$, or just ${\mathfrak{s}u}(V)$, consists of those linear transformations in ${\mathfrak{u}}(V,B)$ with trace zero.

Title (special) unitary Lie algebra specialUnitaryLieAlgebra 2013-03-22 18:45:21 2013-03-22 18:45:21 pan (17366) pan (17366) 6 pan (17366) Definition msc 17B99 unitary algebra special unitary algebra unitary algebra special unitary algebra