Poisson–Lie group

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.

A Poisson–Lie group is a Lie group ${\displaystyle G}$ equipped with a Poisson bracket for which the group multiplication ${\displaystyle \mu :G\times G\to G}$ with ${\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}}$ is a Poisson map, where the manifold ${\displaystyle G\times G}$ has been given the structure of a product Poisson manifold.

Explicitly, the following identity must hold for a Poisson–Lie group:

${\displaystyle \{f_{1},f_{2}\}(gg')=\{f_{1}\circ L_{g},f_{2}\circ L_{g}\}(g')+\{f_{1}\circ R_{g^{\prime }},f_{2}\circ R_{g'}\}(g)}$

where ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are real-valued, smooth functions on the Lie group, while ${\displaystyle g}$ and ${\displaystyle g'}$ are elements of the Lie group. Here, ${\displaystyle L_{g}}$ denotes left-multiplication and ${\displaystyle R_{g}}$ denotes right-multiplication.

If ${\displaystyle {\mathcal {P}}}$ denotes the corresponding Poisson bivector on ${\displaystyle G}$, the condition above can be equivalently stated as

${\displaystyle {\mathcal {P}}(gg')=L_{g\ast }({\mathcal {P}}(g'))+R_{g'\ast }({\mathcal {P}}(g))}$

In particular, taking ${\displaystyle g=g'=e}$ one obtains ${\displaystyle {\mathcal {P}}(e)=0}$, or equivalently ${\displaystyle \{f,g\}(e)=0}$. Applying Weinstein splitting theorem to ${\displaystyle e}$ one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

The Lie algebra ${\displaystyle {\mathfrak {g}}}$ of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor ${\displaystyle P:G\to TG\wedge TG}$ at the identity, i.e. ${\textstyle \delta :=d_{e}P:{\mathfrak {g}}\to {\mathfrak {g}}\wedge {\mathfrak {g}}}$ is a comultiplication. Moreover, the algebra and the coalgebra structure are compatible, i.e. ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle G}$ has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual ${\displaystyle {\mathfrak {g}}^{*}}$ of its bialgebra.^[1][2][3]

A Poisson–Lie group homomorphism ${\displaystyle \phi :G\to H}$ 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 ${\displaystyle \iota :G\to G}$ taking ${\displaystyle \iota (g)=g^{-1}}$ is not a Poisson map either, although it is an anti-Poisson map:

${\displaystyle \{f_{1}\circ \iota ,f_{2}\circ \iota \}=-\{f_{1},f_{2}\}\circ \iota }$

for any two smooth functions ${\displaystyle f_{1},f_{2}}$ on ${\displaystyle G}$.

• Any trivial Poisson structure on a Lie group ${\displaystyle G}$ defines a Poisson–Lie group structure, whose bialgebra is simply ${\displaystyle {\mathfrak {g}}}$ with the trivial comultiplication.
• The dual ${\displaystyle {\mathfrak {g}}^{*}}$ 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.

Let ${\displaystyle G}$ be any semisimple Lie group. Choose a maximal torus ${\displaystyle T\subset G}$ and a choice of positive roots. Let ${\displaystyle B_{\pm }\subset G}$ be the corresponding opposite Borel subgroups, so that ${\displaystyle T=B_{-}\cap B_{+}}$ and there is a natural projection ${\displaystyle \pi :B_{\pm }\to T}$. Then define a Lie group

${\displaystyle G^{*}:=\{(g_{-},g_{+})\in B_{-}\times B_{+}\ {\bigl \vert }\ \pi (g_{-})\pi (g_{+})=1\}}$

which is a subgroup of the product ${\displaystyle B_{-}\times B_{+}}$, and has the same dimension as ${\displaystyle G}$.

The standard Poisson–Lie group structure on ${\displaystyle G}$ is determined by identifying the Lie algebra of ${\displaystyle G^{*}}$ with the dual of the Lie algebra of ${\displaystyle G}$, as in the standard Lie bialgebra example. This defines a Poisson–Lie group structure on both ${\displaystyle G}$ and on the dual Poisson Lie group ${\displaystyle G^{*}}$. This is the "standard" example: the Drinfeld-Jimbo quantum group ${\displaystyle U_{q}{\mathfrak {g}}}$ is a quantization of the Poisson algebra of functions on the group ${\displaystyle G^{*}}$. Note that ${\displaystyle G^{*}}$ is solvable, whereas ${\displaystyle G}$ is semisimple.

• Lie bialgebra
• Quantum group
• Affine quantum group
• Quantum affine algebras

• Doebner, H.-D.; Hennig, J.-D., eds. (1989). Quantum groups. Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG. Berlin: Springer-Verlag. ISBN 3-540-53503-9.
• Chari, Vyjayanthi; Pressley, Andrew (1994). A Guide to Quantum Groups. Cambridge: Cambridge University Press Ltd. ISBN 0-521-55884-0.