Manin Triple

In mathematics, a Manin triple $(\mathfrak, \mathfrak, \mathfrak)$ consists of a Lie algebra ''$\mathfrak$'' with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras ''$\mathfrak$'' and ''$\mathfrak$'' such that ''$\mathfrak$'' is the direct sum of ''$\mathfrak$'' and ''$\mathfrak$'' 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 ''$\mathfrak$'' 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 $(\mathfrak, \mathfrak, \mathfrak)$ is a finite-dimensional Manin triple, then ''$\mathfrak$'' can be made into a Lie bialgebra by letting the cocommutator map $\mathfrak \to \mathfrak \otimes \mathfrak$ be the dual of the Lie bracket $\mathfrak \otimes \mathfrak \to \mathfrak$ (using the fact that the symmetric bilinear form on ''$\mathfrak$'' identifies ''$\mathfrak$'' with the dual of ''$\mathfrak$'').

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

Examples

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

References

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Lie algebras