Abstract
If A is a k-algebra, where k is a commutative ring, then Kassel and w x Ž . w Ž . Ž .x Ž . Loday in 5 study sl A s gl A , gl A , where gl A is the Lie n n n n algebra of n = n matrices over A. They define a Steinberg Lie algebra Ž . st A , which they prove is a universal central Lie algebra extension of n Ž . sl A . Furthermore the kernel of this extension is isomorphic to a cyclic n Ž . homology group, which in our current notation is designated as HC A . 1 In this paper we prove a similar result for A, a k-algebra with involution, a a a, which acts trivially on k. Given a form parameter, L, and Ž . Ž . Ž l g center A , such that ll s 1, we have a form algebra A, L, l see w x. 1 . The Lie algebra adaptation of the general L-quadratic group Ž . Ž . GQ A, L will be denoted as gq L and will be a sub-Lie algebra of 2 n 2 n Ž . Ž . gl A . We will show that the commutator subalgebra of gq L does not 2 n 2 n depend on the choice of parameter L. This subalgebra will be denoted as Ž . sq A and will be called the special quadratic Lie algebra of A. There is 2 n Ž . another subalgebra, sq L , which does depend on the choice of L. We 2 n
Published Version
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