Manin triple

Template:Short description In mathematics, a Manin triple $(𝔤,𝔭,𝔮)$ consists of a Lie algebra $𝔤$ with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras $𝔭$ and $𝔮$ such that $𝔤$ is the direct sum of $𝔭$ and $𝔮$ 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 Template:Interlanguage link classified Manin triples where $𝔤$ 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 $(𝔤,𝔭,𝔮)$ is a finite-dimensional Manin triple, then $𝔭$ can be made into a Lie bialgebra by letting the cocommutator map $𝔭\to 𝔭\otimes 𝔭$ be the dual of the Lie bracket $𝔮\otimes 𝔮\to 𝔮$ (using the fact that the symmetric bilinear form on $𝔤$ identifies $𝔮$ with the dual of $𝔭$).

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

Examples

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