Quasi-Hopf algebra
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. A quasi-Hopf algebra is a quasi-bialgebra B A = ( A , Δ , ε , Φ ) {\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )} for which there exist α , β ∈ A {\displaystyle \alpha ,\beta \in {\mathcal {A}}} and a bijective antihomomorphism S (antipode) of A {\displaystyle {\mathcal {A}}} such that ∑ i S ( b i ) α c i = ε ( a ) α {\displaystyle \sum _{i}S(b_{i})\alpha c_{i}=\varepsilon (a)\alpha } ∑ i b i β S ( c i ) = ε ( a ) β {\displaystyle \sum _{i}b_{i}\beta S(c_{i})=\varepsilon (a)\beta } for all a ∈ A {\displaystyle a\in {\mathcal {A}}} and where Δ ( a ) = ∑ i b i ⊗ c i {\displaystyle \Delta (a)=\sum _{i}b_{i}\otimes c_{i}} and ∑ i X i β S ( Y i ) α Z i = I , {\displaystyle \sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb {I} ,} ∑ j S ( P j ) α Q j β S ( R j ) = I .