Manin triple

Short description: Mathematics concept

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 $\mathfrak{𝔥}$.