Categories of Jordan Structures and Graded Lie Algebras

Abstract

In the paper we describe the subcategory of the category of Zgraded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the 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 H gr 2 (L) 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.

On the Free Jordan Algebras

“…The space SI 8 (3) contains the special identity G 8 , discovered in [6], which is called the Glennie's identity. It is multi-homogenous of degree (3,3,2). In addition of the original expression, there are two simpler formulas due to Thedy and Shestakov [16] and [23].…”

Universal imbedding of a Hom-Lie Triple System

“…for D ∈ HDR(T ) and a, b ∈ T . Furthermore, formula (8) and identity (10) imply that the map λ : T ∧ T → HDR(T ) defined by λ(a ∧ b) = D ab is a module homomorphism from T ∧ T to the adjoint module HDR(T ) and that…”

“…Ternary algebras, that is algebras with ternary multiplication are studied heavily in Lie and Jordan theories, geometry, analysis and physics. For example, Jordan triple system ( [7], [10], [11], [13]) can be realized as 3-graded Lie algebras through the TKK construction ( [8]), from which special Lie algebras can be obtained. Also more to the heart of this study are Lie triple systems, which give rise to Z 2graded Lie algebras (see [6], [9]), which are Lie algebras associated to symmetric spaces.…”

Fine gradings on simple exceptional Jordan pairs and triple systems

“…Consider the 3-graded Lie algebra L = TKK(V) = L −1 ⊕ L 0 ⊕ L 1 defined by the TKK construction, due to Tits, Kantor and Koecher (see [CS11] and references therein). That is,…”

“…Let V be a Jordan pair. Recall that the inner derivations are defined by ν(x, y) := (D(x, y), −D(y, x)) ∈ gl(V + )⊕gl(V − ), where (x, y) ∈ V. Consider the 3-graded Lie algebra L = TKK(V) = L −1 ⊕ L 0 ⊕ L 1 defined by the TKK construction, due to Tits, Kantor and Koecher (see [CS11] and references therein). That is,…”