Quasi-bialgebra

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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[edit]

A quasi-bialgebra \mathcal{B_A} = (\mathcal{A}, \Delta, \varepsilon, \Phi,l,r) is an algebra \mathcal{A} over a field \mathbb{F} equipped with morphisms of algebras

\Delta : \mathcal{A} \rightarrow \mathcal{A \otimes A}
\varepsilon : \mathcal{A} \rightarrow \mathbb{F}

along with invertible elements \Phi \in \mathcal{A \otimes A \otimes A}, and r,l \in A such that the following identities hold:

(id \otimes \Delta) \circ \Delta(a) = \Phi \lbrack (\Delta \otimes id) \circ \Delta (a) \rbrack \Phi^{-1}, \quad \forall a \in \mathcal{A}
\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)
(\varepsilon \otimes id)(\Delta a) = l^{-1} a l, \qquad (id \otimes \varepsilon) \circ \Delta = r^{-1} a r, \quad \forall a \in \mathcal{A}
(id \otimes \varepsilon \otimes id)(\Phi) = 1 \otimes 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 \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[2]. 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[3]. Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: l=r=1 and \Phi=1 \otimes 1 \otimes 1.

Braided Quasi-Bialgebras[edit]

A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category \mathcal{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 (\mathcal{A},\Delta,\epsilon,\Phi,l,r) is braided if it has a universal R-matrix, ie an invertible element R \in \mathcal{A \otimes A} such that the following 3 identities hold:

(\Delta^{op})(a)=R \Delta(a) R^{-1}
(id \otimes \Delta)(R)=(\Phi_{231})^{-1} R_{13} \Phi_{213} R_{12} (\Phi_{213})^{-1}
(\Delta \otimes id)(R)=(\Phi_{321}) R_{13} (\Phi_{213})^{-1} R_{23} \Phi_{123}

Where, for every a_1 \otimes ... \otimes a_k \in \mathcal{A}^{\otimes k}, a_{i_1 i_2 ... i_n} is the monomial with a_j in the i_jth spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of \mathcal{A}^{\otimes k}[4].

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}[5]

Twisting[edit]

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume r=l=1) .

If \mathcal{B_A} is a quasi-bialgebra and F \in \mathcal{A \otimes A} is an invertible element such that (\varepsilon \otimes id) F = (id \otimes \varepsilon) F = 1 , set

 \Delta ' (a) = F \Delta (a) F^{-1}, \quad \forall a \in \mathcal{A}
 \Phi ' = (1 \otimes  F) \ ((id \otimes \Delta) F) \ \Phi \ ((\Delta \otimes id)F^{-1}) \ (F^{-1} \otimes 1).

Then, the set (\mathcal{A}, \Delta ' , \varepsilon, \Phi ') is also a quasi-bialgebra obtained by twisting  \mathcal{B_A} by F, which is called a twist or gauge transformation[6]. If (\mathcal{A},\Delta,\varepsilon, \Phi) was a braided quasi-bialgebra with universal R-matrix R , then so is (\mathcal{A},\Delta',\varepsilon, \Phi ') with universal R-matrix F_{21}RF^{-1} (using the notation from the above section)[7]. 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 \mathcal{B_A}, \mathcal{B_{A'}} be quasi-bialgebras, let \mathcal{B'_{A'}} be the twisting of \mathcal{B_{A'}} by F, and let there exist an isomorphism: \alpha:\mathcal{B_A} \to \mathcal{B'_{A'}}. Then the induced tensor functor (\alpha^*,id,\phi_2^F) is a tensor category equivalence between \mathcal{A'}-mod and \mathcal{A}-mod. Where \phi_2^F(v \otimes w)=F^{-1}(v \otimes w). Moreover, if \alpha is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence[8].

Usage[edit]

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 Heisenberg XXZ model in the framework of the Algebraic Bethe ansatz.

See Also[edit]

References[edit]

  1. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 369 Springer-Verlag. ISBN 0387943706
  2. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 368 Springer-Verlag. ISBN 0387943706
  3. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 370 Springer-Verlag. ISBN 0387943706
  4. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 371 Springer-Verlag. ISBN 0387943706
  5. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 372 Springer-Verlag. ISBN 0387943706
  6. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 373 Springer-Verlag. ISBN 0387943706
  7. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 376 Springer-Verlag. ISBN 0387943706
  8. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 375,376 Springer-Verlag. ISBN 0387943706

Further Reading[edit]

  • 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