In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold.
The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups.
equipped with a Poisson bracket for which the group multiplication
is a Poisson map, where the manifold
has been given the structure of a product Poisson manifold.
Explicitly, the following identity must hold for a Poisson–Lie group: where
are real-valued, smooth functions on the Lie group, while
denotes the corresponding Poisson bivector on
, the condition above can be equivalently stated as In particular, taking
Applying Weinstein splitting theorem to
one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.
of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor
Moreover, the algebra and the coalgebra structure are compatible, i.e.
is a Lie bialgebra, The classical Lie group–Lie algebra correspondence, which gives an equivalence of categories between simply connected Lie groups and finite-dimensional Lie algebras, was extended by Drinfeld to an equivalence of categories between simply connected Poisson–Lie groups and finite-dimensional Lie bialgebras.
Thanks to Drinfeld theorem, any Poisson–Lie group
is defined to be both a Lie group homomorphism and a Poisson map.
These two example are dual of each other via Drinfeld theorem, in the sense explained above.
be any semisimple Lie group.
Choose a maximal torus
and a choice of positive roots.
be the corresponding opposite Borel subgroups, so that
Then define a Lie group which is a subgroup of the product
The standard Poisson–Lie group structure on
is determined by identifying the Lie algebra of
with the dual of the Lie algebra of
, as in the standard Lie bialgebra example.
This defines a Poisson–Lie group structure on both
and on the dual Poisson Lie group
This is the "standard" example: the Drinfeld-Jimbo quantum group
is a quantization of the Poisson algebra of functions on the group