Poisson–Lie group explained
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.
Many quantum groups are quantizations of the Poisson algebra of functions on a Poisson–Lie group.
Definition
A Poisson–Lie group is a Lie group
equipped with a Poisson bracket for which the group multiplication
with
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:
\{f1,f2\}(gg')=\{f1\circLg,f2\circLg\}(g')+\{f1\circ
,f2\circRg'\}(g)
where
and
are real-valued, smooth functions on the Lie group, while
and
are elements of the Lie group. Here,
denotes left-multiplication and
denotes right-multiplication.
If
denotes the corresponding Poisson bivector on
, the condition above can be equivalently stated as
l{P}(gg')=Lg(l{P}(g'))+Rg'(l{P}(g))
In particular, taking
one obtains
, or equivalently
. Applying Weinstein splitting theorem to
one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.Poisson-Lie groups - Lie bialgebra correspondence
The Lie algebra
of a Poisson–Lie group has a natural structure of
Lie coalgebra given by linearising the Poisson tensor
at the identity, i.e.
is a
comultiplication. 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
has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual
of its bialgebra.[1] [2] [3] Homomorphisms
A Poisson–Lie group homomorphism
is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, neither left translations nor right translations are Poisson maps. Also, the inversion map
taking
is not a Poisson map either, although it is an anti-Poisson map:
\{f1\circ\iota,f2\circ\iota\}=-\{f1,f2\}\circ\iota
for any two smooth functions
on
.
Examples
Trivial examples
- Any trivial Poisson structure on a Lie group
defines a Poisson–Lie group structure, whose bialgebra is simply
with the trivial comultiplication.
of a Lie algebra, together with its linear Poisson structure, is an additive Poisson–Lie group.These two example are dual of each other via Drinfeld theorem, in the sense explained above.
Other examples
Let
be any
semisimple Lie group. Choose a maximal torus
and a choice of
positive roots. Let
be the corresponding opposite
Borel subgroups, so that
and there is a natural projection
.Then define a Lie group
G*:=\{(g-,g+)\inB- x B+ l\vert \pi(g-)\pi(g+)=1\}
which is a subgroup of the product
, and has the same dimension as
.
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
.Note that
is solvable, whereas
is semisimple.See also
References
- Book: H.-D. . Doebner . J.-D. . Hennig . Quantum groups . Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG . 1989 . Springer-Verlag . Berlin . 3-540-53503-9 .
- Book: Vyjayanthi . Chari. Vyjayanthi Chari . Andrew . Pressley . A Guide to Quantum Groups . registration . 1994 . Cambridge University Press Ltd. . Cambridge . 0-521-55884-0 .
Notes and References
- Lu . Jiang-Hua . Weinstein . Alan . Alan Weinstein . 1990-01-01 . Poisson Lie groups, dressing transformations, and Bruhat decompositions . Journal of Differential Geometry . 31 . 2 . 10.4310/jdg/1214444324 . 117053536 . 0022-040X. free .
- Kosmann-Schwarzbach . Y. . Yvette Kosmann-Schwarzbach . 1996-12-01 . Poisson-Lie groups and beyond . Journal of Mathematical Sciences . en . 82 . 6 . 3807–3813 . 10.1007/BF02362640 . 123117926 . 1573-8795.
- Proceedings of the International Center for Pure and Applied Mathematics at Pondicherry University, 8–26 January 1996. Kosmann-Schwarzbach . Y. . Lie bialgebras, poisson Lie groups and dressing transformations . Yvette Kosmann-Schwarzbach . 1997 . Integrability of Nonlinear Systems . https://link.springer.com/chapter/10.1007/BFb0113695 . Lecture Notes in Physics . 495 . Y. Kosmann-Schwarzbach. B. Grammaticos. K. M. Tamizhmani . en . Berlin, Heidelberg . Springer . 104–170 . 10.1007/BFb0113695 . 978-3-540-69521-9.