(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
Canonical name	specialUnitaryLieAlgebra
Date of creation	2013-03-22 18:45:21
Last modified on	2013-03-22 18:45:21
Owner	pan (17366)
Last modified by	pan (17366)
Numerical id	6
Author	pan (17366)
Entry type	Definition
Classification	msc 17B99
Synonym	unitary algebra
Synonym	special unitary algebra
Defines	unitary algebra
Defines	special unitary algebra