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 Lie algebra of a Poisson–Lie group is a Lie bialgebra. 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 ${\displaystyle \mu :G\times G\to G}$ with ${\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}}$ is a Poisson map, where the manifold G×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 f₁ and f₂ 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 ${\displaystyle {\mathcal {P}}}$ denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as

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

Note that for Poisson-Lie group always ${\displaystyle \{f,g\}(e)=0}$, or equivalently ${\displaystyle {\mathcal {P}}(e)=0}$. This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

Example[]

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 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.

Homomorphisms[]

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 G.

See also[]

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

References[]

• 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.