Quasi-bialgebra explained

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

\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

l{B _{A}} = ( l{A}, \Delta, \varepsilon, \Phi,l,r) is an algebra l{A} over a field F equipped with morphisms of algebras\Delta : l{A} → l{A ⊗ A} \varepsilon : l{A} → F

along with invertible elements

\Phi \in l{A ⊗ A ⊗ A} , and r,l \in A such that the following identities hold:(id ⊗ \Delta) \circ \Delta(a) = \Phi \lbrack (\Delta ⊗ id) \circ \Delta (a) \rbrack \Phi ^{-1} , \forall a \in l{A} \lbrack (id ⊗ id ⊗ \Delta)(\Phi) \rbrack \lbrack (\Delta ⊗ id ⊗ id)(\Phi) \rbrack = ( 1 ⊗ \Phi) \lbrack (id ⊗ \Delta ⊗ id)(\Phi) \rbrack (\Phi ⊗ 1 ) (\varepsilon ⊗ id)(\Delta a) = l ^{-1} a l, (id ⊗ \varepsilon) \circ \Delta = r ^{-1} a r, \forall a \in l{A} (id ⊗ \varepsilon ⊗ id)(\Phi) = r ⊗ l ^{-1} .

Where

\Delta and \epsilon are called the comultiplication and counit, r and l are called the right and left unit constraints (resp.), and \Phi is sometimes called the Drinfeld associator.^[1] This definition is constructed so that the category l{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.^[1] Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. l = r =1 the definition may sometimes be given with this assumed.^[1] Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: l = r =1 and \Phi =1 ⊗ 1 ⊗ 1 .

Braided quasi-bialgebras

A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category

l{A}-Mod is braided. Equivalently, by analogy with braided bialgebras, we can construct a notion of a universal R-matrix which controls the non-cocommutativity of a quasi-bialgebra. The definition is the same as in the braided bialgebra case except for additional complications in the formulas caused by adding in the associator.

Proposition: A quasi-bialgebra

( l{A},\Delta,\epsilon,\Phi,l,r) is braided if it has a universal R-matrix, ie an invertible element R \in l{A ⊗ A} such that the following 3 identities hold:(\Delta ^{op} )(a) = R \Delta(a) R ^{-1} (id ⊗ \Delta)(R) = (\Phi _{231} ) ^{-1} R _{13} \Phi _{213} R _{12} (\Phi _{213} ) ^{-1} (\Delta ⊗ id)(R) = (\Phi _{321} ) R _{13} (\Phi _{213} ) ^{-1} R _{23} \Phi _{123}

Where, for every

a _{1} ⊗ ... ⊗ a _{k} \in l{A} ^{⊗} , a _{i _{1} i _{2} ... i _{n}} is the monomial with a _{j} in the i _{j} th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of l{A} ^{⊗} .^[1]

Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang–Baxter equation:

R _{12} \Phi _{321} R _{13} (\Phi _{132} ) ^{-1} R _{23} \Phi _{123} = \Phi _{321} R _{23} (\Phi _{231} ) ^{-1} R _{13} \Phi _{213} R _{12} ^[1]

Twisting

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume

r = l =1 ) .

If

l{B _{A}} is a quasi-bialgebra and F \in l{A ⊗ A} is an invertible element such that (\varepsilon ⊗ id) F = (id ⊗ \varepsilon) F = 1 , set\Delta ' (a) = F \Delta (a) F ^{-1} , \forall a \in l{A} \Phi ' = ( 1 ⊗ F) ((id ⊗ \Delta) F) \Phi ((\Delta ⊗ id)F ^{-1} ) (F ^{-1} ⊗ 1 ).

Then, the set

( l{A}, \Delta ' , \varepsilon, \Phi ') is also a quasi-bialgebra obtained by twisting l{B _{A}} by F, which is called a twist or gauge transformation.^[1] If ( l{A},\Delta,\varepsilon, \Phi) was a braided quasi-bialgebra with universal R-matrix R , then so is ( l{A},\Delta',\varepsilon, \Phi ') with universal R-matrix F _{21} RF ^{-1} (using the notation from the above section).^[1] However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by F _{1} and then F _{2} is equivalent to twisting by F _{2F} _{1} , and twisting by F then F ^{-1} recovers the original quasi-bialgebra.

Twistings have the important property that they induce categorical equivalences on the tensor category of modules:

Theorem: Let

l{B _{A}} , l{B _{A'} } be quasi-bialgebras, let l{B' _{A'} } be the twisting of l{B _{A'} } by F , and let there exist an isomorphism: \alpha: l{B _{A}} \to l{B' _{A'} }. Then the induced tensor functor (\alpha ^{F)} _{2} is a tensor category equivalence between l{A'}-mod and l{A}-mod . Where \phi ^{F(v} _{2} ⊗ w) = F ^{-1} (v ⊗ w) . Moreover, if \alpha is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.^[1]

Usage

Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the XXZ in the framework of the Algebraic Bethe ansatz.

See also

• Bialgebra
• Hopf algebra
• Quasi-Hopf algebra

References

1. C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Springer-Verlag.

Further reading

• Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457
• J.M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000