Manin triple

 ( Lie Algebras ) 

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

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

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

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

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Examples

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

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