Manin triple

In mathematics, a Manin triple $({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})$ consists of a Lie algebra ${\mathfrak {g}}$ with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras ${\mathfrak {p}}$ and ${\mathfrak {q}}$ such that ${\mathfrak {g}}$ is the direct sum of ${\mathfrak {p}}$ and ${\mathfrak {q}}$ 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.

In 2001 Delorme classified Manin triples where ${\mathfrak {g}}$ is a complex reductive Lie algebra.

Manin triples and Lie bialgebras

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

More precisely, if $({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})$ is a finite-dimensional Manin triple, then ${\mathfrak {p}}$ can be made into a Lie bialgebra by letting the cocommutator map ${\mathfrak {p}}\to {\mathfrak {p}}\otimes {\mathfrak {p}}$ be the dual of the Lie bracket ${\mathfrak {q}}\otimes {\mathfrak {q}}\to {\mathfrak {q}}$ (using the fact that the symmetric bilinear form on ${\mathfrak {g}}$ identifies ${\mathfrak {q}}$ with the dual of ${\mathfrak {p}}$).

Conversely if ${\mathfrak {p}}$ is a Lie bialgebra then one can construct a Manin triple $({\mathfrak {p}}\oplus {\mathfrak {p}}^{*},{\mathfrak {p}},{\mathfrak {p}}^{*})$ by letting ${\mathfrak {q}}$ be the dual of ${\mathfrak {p}}$ and defining the commutator of ${\mathfrak {p}}$ and ${\mathfrak {q}}$ to make the bilinear form on ${\mathfrak {g}}={\mathfrak {p}}\oplus {\mathfrak {q}}$ invariant.

Examples

• Suppose that ${\mathfrak {a}}$ is a complex semisimple Lie algebra with invariant symmetric bilinear form $(\cdot ,\cdot )$. Then there is a Manin triple $({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})$ with ${\mathfrak {g}}={\mathfrak {a}}\oplus {\mathfrak {a}}$, with the scalar product on ${\mathfrak {g}}$ given by $((w,x),(y,z))=(w,y)-(x,z)$. The subalgebra ${\mathfrak {p}}$ is the space of diagonal elements $(x,x)$, and the subalgebra ${\mathfrak {q}}$ is the space of elements $(x,y)$ with $x$ in a fixed Borel subalgebra containing a Cartan subalgebra ${\mathfrak {h}}$, $y$ in the opposite Borel subalgebra, and where $x$ and $y$ have the same component in ${\mathfrak {h}}$.