In mathematics, a Manin triple (''g'', ''p'', ''q'') consists of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
''g'' with a non-degenerate invariant
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
, 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 , who named them after
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logi ...
.
classified the Manin triples where ''g'' is a complex
reductive Lie algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfr ...
.
Manin triples and Lie bialgebras
If (''g'', ''p'', ''q'') is a finite-dimensional Manin triple then ''p'' can be made into a
Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi ...
by letting the
cocommutator map In mathematics a Lie coalgebra is the dual structure to a Lie algebra.
In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely.
Definition
Let ''E'' be a v ...
''p'' → ''p'' ⊗ ''p'' be dual to the map ''q'' ⊗ ''q'' → ''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 from it by letting ''q'' be the dual of ''p'' and defining the commutator of ''p'' and ''q'' to make the bilinear form on ''g'' = ''p'' ⊕ ''q'' invariant.
Examples
*Suppose that ''a'' is a complex semisimple Lie algebra with invariant symmetric bilinear form (,). Then there is a Manin triple (''g'',''p'',''q'') with ''g'' = ''a''⊕''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 In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel.
If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, ...
containing a Cartan subalgebra ''h'', ''y'' in the opposite Borel subalgebra, and where ''x'' and ''y'' have the same component in ''h''.
References
*
*{{Citation , last1=Drinfeld , first1=V. G. , title=Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986) , chapter-url=http://www.mathunion.org/ICM/ICM1986.1/ , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , isbn= 978-0-8218-0110-9 , mr=934283 , year=1987 , volume=1 , chapter=Quantum groups , pages=798–820
Lie algebras