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 
  
    
      
        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:
 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
 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.
 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.
 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:
 for any two smooth functions 
  
    
      
        
          f
          
            1
          
        
        ,
        
          f
          
            2
          
        
      
    
    {\displaystyle f_{1},f_{2}}
  
 on 
  
    
      
        G
      
    
    {\displaystyle G}
  
.
 These two example are dual of each other via Drinfeld theorem, in the sense explained above.
 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 
 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.