Poisson–Lie group

Poisson manifold that is also a 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.

## Definition

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A Poisson–Lie group is a Lie group  G {\displaystyle G} equipped with a Poisson bracket for which the group multiplication  μ : G × G -> G {\displaystyle \mu :G\times G\to G} with  μ ( g 1 , g 2 ) = g 1 g 2 {\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}} is a Poisson map, where the manifold  G × G {\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:

{ f 1 , f 2 } ( g g ′ ) = { f 1 ∘ L g , f 2 ∘ L g } ( g ′ ) + { f 1 ∘ R g ′ , f 2 ∘ R g ′ } ( g ) {\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  f 1 {\displaystyle f_{1}} and  f 2 {\displaystyle f_{2}} are real-valued, smooth functions on the Lie group, while  g {\displaystyle g} and  g ′ {\displaystyle g'} are elements of the Lie group. Here,  L g {\displaystyle L_{g}} denotes left-multiplication and  R g {\displaystyle R_{g}} denotes right-multiplication.

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

P ( g g ′ ) = L g ∗ ( P ( g ′ ) ) + R g ′ ∗ ( P ( g ) ) {\displaystyle {\mathcal {P}}(gg')=L_{g\ast }({\mathcal {P}}(g'))+R_{g'\ast }({\mathcal {P}}(g))}

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

## Poisson-Lie groups - Lie bialgebra correspondence

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The Lie algebra  g {\displaystyle {\mathfrak {g}}} of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor  P : G -> T G ∧ T G {\displaystyle P:G\to TG\wedge TG} at the identity, i.e.  δ := d e P : g -> g ∧ g {\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.  g {\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  G {\displaystyle G} has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual  g ∗ {\displaystyle {\mathfrak {g}}^{*}} of its bialgebra.[1][2][3]

## Homomorphisms

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

{ f 1 ∘ ι , f 2 ∘ ι } = − { f 1 , f 2 } ∘ ι {\displaystyle \\{f_{1}\circ \iota ,f_{2}\circ \iota \\}=-\\{f_{1},f_{2}\\}\circ \iota }

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

## Examples

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### Trivial examples

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

### Other examples

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

G ∗ := { ( g − , g + ) ∈ B − × B + | π ( g − ) π ( g + ) = 1 } {\displaystyle G^{*}:=\\{(g_{-},g_{+})\in B_{-}\times B_{+}\ {\bigl \vert }\ \pi (g_{-})\pi (g_{+})=1\\}}

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

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

## See also

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

## References

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1. ^ Lu, Jiang-Hua; Weinstein, Alan (1990-01-01). "Poisson Lie groups, dressing transformations, and Bruhat decompositions". Journal of Differential Geometry. 31 (2). doi:10.4310/jdg/1214444324. ISSN 0022-040X. S2CID 117053536.
2. ^ Kosmann-Schwarzbach, Y. (1996-12-01). "Poisson-Lie groups and beyond". Journal of Mathematical Sciences. 82 (6): 3807–3813. doi:10.1007/BF02362640. ISSN 1573-8795. S2CID 123117926.
3. ^ Kosmann-Schwarzbach, Y. (1997). "Lie bialgebras, poisson Lie groups and dressing transformations". In Y. Kosmann-Schwarzbach; B. Grammaticos; K. M. Tamizhmani (eds.). Integrability of Nonlinear Systems. Proceedings of the International Center for Pure and Applied Mathematics at Pondicherry University, 8–26 January 1996. Lecture Notes in Physics. Vol. 495\. Berlin, Heidelberg: Springer. pp. 104–170. doi:10.1007/BFb0113695. ISBN 978-3-540-69521-9.

* 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.