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.