Categories of Jordan Structures and Graded Lie Algebras

Abstract

In the paper we describe the subcategory of the category of ℤ-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the Tits, Kantor, and Koecher TKK construction. For instance, we prove that L = L −1 ⊕ L 0 ⊕ L 1 can be constructed from a Jordan pair if and only if L 0 = [L −1, L 1] and the second graded homology group is trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras.