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 ' and ' such that ' is the direct sum of ' and ' 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.

In 2001 classified Manin triples where ' is a complex reductive Lie algebra.

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 ' with the dual of ').

Conversely if ' is a Lie bialgebra then one can construct a Manin triple by letting ' be the dual of ' and defining the commutator of ' and ' to make the bilinear form on ' invariant.

• Suppose that ' is a complex semisimple Lie algebra with invariant symmetric bilinear form '. Then there is a Manin triple with , with the scalar product on ' given by '. The subalgebra ' is the space of diagonal elements ', and the subalgebra ' is the space of elements ' with ' in a fixed Borel subalgebra containing a Cartan subalgebra ', ' in the opposite Borel subalgebra, and where ' and ' have the same component in '.