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From Wikipedia, the free encyclopedia
In mathematics, a Manin triple (g, p, q) consists of a Lie algebra g with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p and q such that g is the direct sum of p and q as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition.
Manin triples were introduced by Drinfeld (1987, p.802), who named them after Yuri Manin.
Delorme (2001) classified the Manin triples where g is a complex reductive Lie algebra.
If (g, p, q) is a finite-dimensional Manin triple then p can be made into a Lie bialgebra by letting the cocommutator map p → p ⊗ p be dual to the map q ⊗ q → q (using the fact that the symmetric bilinear form on g identifies q with the dual of p).
Conversely if p is a Lie bialgebra then one can construct a Manin triple from it by letting q be the dual of p and defining the commutator of p and q to make the bilinear form on g = p ⊕ q invariant.