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
Famous quotes containing the words jordan and/or pair:
“To rescue our children we will have to let them save us from the power we embody: we will have to trust the very difference that they forever personify. And we will have to allow them the choice, without fear of death: that they may come and do likewise or that they may come and that we will follow them, that a little child will lead us back to the child we will always be, vulnerable and wanting and hurting for love and for beauty.”
—June Jordan (b. 1939)
“I should have been a pair of ragged claws
Scuttling across the floors of silent seas.”
—T.S. (Thomas Stearns)