Poisson Lie group


Group Theory

Symplectic geometry



A Poisson Lie group (often written Poisson-Lie group) or Lie Poisson group GG is a Lie group and a Poisson manifolds, the two structures being compatible such that the group product is a smooth map of Poisson manifolds where G×GG\times G has the product Poisson structure.

Warning: the inverse map is NOT a Poisson map unless G has the trivial Poisson structure, in fact it is an anti-Poisson map.

Deformation quantizations of Poisson Lie groups are Hopf algebras. The usual quantum groups have smaller number of quantum subgroups (i.e. Hopf quotient algebras) than the corresponding Lie group has, namely only only those whose classical limits are not only Lie subgroups but Poisson Lie subgroups.


Relation to quantum groups

One can regard Poisson groups as the classical limit of quantum groups: a theorem by Drinfeld established a bijection between connected, simply connected Poisson Lie groups and Lie bialgebras.


Additive Poisson Lie groups

A Lie-Poisson structure is an additive Poisson Lie group (e.g. Kosmann-Schwarzbach 04, p. 46).

If HH is any (finite dimensional) Lie group then the dual (T eH) *(T_e H)^* of its tangent Lie algebra has a canonical bracket introduced by Kirillov which makes it into a Poisson Lie group. To this aim one identifies (T eH) *(T_e H)^* with its own tangent space T u(T eH) *T_u (T_e H)^* and interprets the differential dfdf of a function f:(T eH) *f:(T_e H)^*\to \mathbb{R} as a function (T eH) *((T eH) *) *T eH(T_e H)^*\to ((T_e H)^*)^*\cong T_e H where the finite dimensionality is used. Then Kirillov defines

{f 1,f 2}(u)=[(df 1) u,(df 2) u,u \lbrace f_1, f_2\rbrace (u) = \langle [(df_1)_u, (df_2)_u, u\rangle

Given two Lie groups H,KH,K, the Lie algebra homomorphisms T eHT eKT_e H \to T_e K are in 1-1 correspondence with the Poisson Lie maps (T eK) *(T eH) *(T_e K)^* \to (T_e H)^*.


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