Manin triple

In mathematics, a Manin triple ${\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})}$ consists of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {q}}}$ such that ${\displaystyle {\mathfrak {g}}}$ is the direct sum of ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {q}}}$ 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 Vladimir Drinfeld in 1987, who named them after Yuri Manin.^[1]

In 2001 Delorme [fr] classified Manin triples where ${\displaystyle {\mathfrak {g}}}$ is a complex reductive Lie algebra.^[2]