Jordan Pair
A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V−. The trilinear form is then replaced by a pair of trilinear forms
which are often viewed as quadratic maps V+ → Hom(V−, V+) and V− → Hom(V+, V−). The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being
and the other being the analogue with + and − subscripts exchanged.
As in the case of Jordan triple systems, one can define, for u in V− and v in V+, a linear map
and similarly L−. The Jordan axioms (apart from symmetry) may then be written
which imply that the images of L+ and L− are closed under commutator brackets in End(V+) and End(V−). Together they determine a linear map
whose image is a Lie subalgebra, and the Jordan identities become Jacobi identities for a graded Lie bracket on
so that conversely, if
is a graded Lie algebra, then the pair is a Jordan pair, with brackets
Jordan triple systems are Jordan pairs with V+ = V− and equal trilinear forms. Another important case occurs when V+ and V− are dual to one another, with dual trilinear forms determined by an element of
These arise in particular when above is semisimple, when the Killing form provides a duality between and .
Read more about this topic: Triple System
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