Manin triple

In mathematics, a Manin triple

consists of a Lie algebra

with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras

is the direct sum of

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 Vladimir Drinfeld in 1987, who named them after Yuri Manin.

[1] In 2001 Delorme [fr] classified Manin triples where

is a complex reductive Lie algebra.

[2] There is an equivalence of categories between finite-dimensional Manin triples and finite-dimensional Lie bialgebras.

More precisely, if

is a finite-dimensional Manin triple, then

can be made into a Lie bialgebra by letting the cocommutator map

be the dual of the Lie bracket

(using the fact that the symmetric bilinear form on

identifies

Conversely if

is a Lie bialgebra then one can construct a Manin triple

and defining the commutator of

to make the bilinear form on

invariant.