(special) unitary Lie algebra

Let V be a vector spaceMathworldPlanetmath over a field K admitting an involutionPlanetmathPlanetmathPlanetmath σ:KK, and let B:V×V𝔽 be a http://planetmath.org/node/SesquilinearFormsOverGeneralFieldshermitian formPlanetmathPlanetmath relative to σ. Then the unitary Lie algebra 𝔲(V,B), or just 𝔲(V), consists of the linear transformations T satisfying


for all x,yV. This is a Lie algebraMathworldPlanetmath over k={αKασ=α}, but not over K in the case that Kk (because B is linear in the first, but not in the second variable).

The special unitary Lie algebra 𝔰u(V,B), or just 𝔰u(V), consists of those linear transformations in 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