On the steinberg lie algebrasst2(R)

摘要

For a nonassociative algebraRover a fieldk, we give a necessary and sufficient condition determining whensl2(R) is a Lie algebra and call suchRsl2-Lie admissible.IfRissl2-Lie admissible and chark ≠2,sl2(R)is the universal covering algebra ofsl2(R), andH2(sl2(R),k) is studied.IfRis associative, we characterizeH2(sl2(R),k)and ProveH2(sl2(R),k) →H2(sl3(R),k) is surjective (not necessarily injective).Also, a class ofsl2-Lie admissible algebras is constructed and as a byproduct, we get a realization of the Heisenberg Lie algebras.