In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element Φ {\displaystyle \Phi } which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.
Definition A quasi-bialgebra B A = ( A , Δ , ε , Φ , l , r ) {\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi ,l,r)} is an algebra A {\displaystyle {\mathcal {A}}} over a field F {\displaystyle \mathbb {F} } equipped with morphisms of algebras
Δ : A → A ⊗ A {\displaystyle \Delta :{\mathcal {A}}\rightarrow {\mathcal {A\otimes A}}}
ε : A → F {\displaystyle \varepsilon :{\mathcal {A}}\rightarrow \mathbb {F} }
along with invertible elements Φ ∈ A ⊗ A ⊗ A {\displaystyle \Phi \in {\mathcal {A\otimes A\otimes A}}} , and r , l ∈ A {\displaystyle r,l\in A} such that the following identities hold:
( i d ⊗ Δ ) ∘ Δ ( a ) = Φ [ ( Δ ⊗ i d ) ∘ Δ ( a ) ] Φ − 1 , ∀ a ∈ A {\displaystyle (id\otimes \Delta )\circ \Delta (a)=\Phi \lbrack (\Delta \otimes id)\circ \Delta (a)\rbrack \Phi ^{-1},\quad \forall a\in {\mathcal {A}}}
[ ( i d ⊗ i d ⊗ Δ ) ( Φ ) ] [ ( Δ ⊗ i d ⊗ i d ) ( Φ ) ] = ( 1 ⊗ Φ ) [ ( i d ⊗ Δ ⊗ i d ) ( Φ ) ] ( Φ ⊗ 1 ) {\displaystyle \lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)}
( ε ⊗ i d ) ( Δ a ) = l − 1 a l , ( i d ⊗ ε ) ∘ Δ = r − 1 a r , ∀ a ∈ A {\displaystyle (\varepsilon \otimes id)(\Delta a)=l^{-1}al,\qquad (id\otimes \varepsilon )\circ \Delta =r^{-1}ar,\quad \forall a\in {\mathcal {A}}}
( i d ⊗ ε ⊗ i d ) ( Φ ) = r ⊗ l − 1 . {\displaystyle (id\otimes \varepsilon \otimes id)(\Phi )=r\otimes l^{-1}.}
Where Δ {\displaystyle \Delta } and ϵ {\displaystyle \epsilon } are called the comultiplication and counit, r {\displaystyle r} and l {\displaystyle l} are called the right and left unit constraints (resp.), and Φ {\displaystyle \Phi } is sometimes called the Drinfeld associator. This definition is constructed so that the category A − M o d {\displaystyle {\mathcal {A}}-Mod} is a tensor category under the usual vector space tensor product, and in fact this can be taken as the definition instead of the list of above identities. Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. l = r = 1 {\displaystyle l=r=1} the definition may sometimes be given with this assumed. Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: l = r = 1 {\displaystyle l=r=1} and Φ = 1 ⊗ 1 ⊗ 1 {\displaystyle \Phi =1\otimes 1\otimes 1} .
… excerpt ends here. Continue reading the full article.
