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]

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

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

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

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