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.

Definition

A Poisson–Lie group is a Lie group $G$ equipped with a Poisson bracket for which the group multiplication $\mu :G\times G\to G$ with $\mu (g_{1},g_{2})=g_{1}g_{2}$ is a Poisson map, where the manifold $G\times G$ 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}\}(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 $f_{1}$ and $f_{2}$ are real-valued, smooth functions on the Lie group, while $g$ and $g'$ are elements of the Lie group. Here, $L_{g}$ denotes left-multiplication and $R_{g}$ denotes right-multiplication.

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

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

In particular, taking $g=g'=e$ one obtains ${\mathcal {P}}(e)=0$ , or equivalently $\{f,g\}(e)=0$ . Applying Weinstein splitting theorem to $e$ 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 ${\mathfrak {g}}$ of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor $P:G\to TG\wedge TG$ at the identity, i.e. ${\textstyle \delta$ is a comultiplication. Moreover, the algebra and the coalgebra structure are compatible, i.e. ${\mathfrak {g}}$ 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 $G$ has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual ${\mathfrak {g}}^{*}$ of its bialgebra.[1][2][3]

Homomorphisms

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

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

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

Examples

Trivial examples

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

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

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

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

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

References

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.
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.
Kosmann-Schwarzbach, Y. (1997). "Lie bialgebras, poisson Lie groups and dressing transformations". Integrability of Nonlinear Systems. Lecture Notes in Physics. Berlin, Heidelberg: Springer. 495: 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.

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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". Integrability of Nonlinear Systems. Lecture Notes in Physics. Berlin, Heidelberg: Springer. 495: 104–170. doi:10.1007/BFb0113695. ISBN 978-3-540-69521-9.