量子群
これらの代数がホップ代数となるような様々な余結合的余積がある。例えば、
Δ 1 ( k λ ) = k λ ⊗ k λ , {\displaystyle \Delta _{1}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda },} Δ 1 ( e i ) = 1 ⊗ e i + e i ⊗ k i , {\displaystyle \Delta _{1}(e_{i})=1\otimes e_{i}+e_{i}\otimes k_{i},} Δ 1 ( f i ) = k i − 1 ⊗ f i + f i ⊗ 1 , {\displaystyle \Delta _{1}(f_{i})=k_{i}^{-1}\otimes f_{i}+f_{i}\otimes 1,} Δ 2 ( k λ ) = k λ ⊗ k λ , {\displaystyle \Delta _{2}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda },} Δ 2 ( e i ) = k i − 1 ⊗ e i + e i ⊗ 1 , {\displaystyle \Delta _{2}(e_{i})=k_{i}^{-1}\otimes e_{i}+e_{i}\otimes 1,} Δ 2 ( f i ) = 1 ⊗ f i + f i ⊗ k i , {\displaystyle \Delta _{2}(f_{i})=1\otimes f_{i}+f_{i}\otimes k_{i},} Δ 3 ( k λ ) = k λ ⊗ k λ , {\displaystyle \Delta _{3}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda },} Δ 3 ( e i ) = k i − 1 2 ⊗ e i + e i ⊗ k i 1 2 , {\displaystyle \Delta _{3}(e_{i})=k_{i}^{-{\frac {1}{2}}}\otimes e_{i}+e_{i}\otimes k_{i}^{\frac {1}{2}},} Δ 3 ( f i ) = k i − 1 2 ⊗ f i + f i ⊗ k i 1 2 , {\displaystyle \Delta _{3}(f_{i})=k_{i}^{-{\frac {1}{2}}}\otimes f_{i}+f_{i}\otimes k_{i}^{\frac {1}{2}},}
Whereas compact matrix pseudogroups are typically versions of Drinfeld–Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first.