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 which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.

Definition

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A quasi-bialgebra   is an algebra   over a field   equipped with morphisms of algebras

 
 

along with invertible elements  , and   such that the following identities hold:

 
 
 
 

Where   and   are called the comultiplication and counit,   and   are called the right and left unit constraints (resp.), and   is sometimes called the Drinfeld associator.[1]: 369–376  This definition is constructed so that the category   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]: 368  Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie.   the definition may sometimes be given with this assumed.[1]: 370  Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints:   and  .

Braided quasi-bialgebras

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A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category   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   is braided if it has a universal R-matrix, ie an invertible element   such that the following 3 identities hold:

 
 
 

Where, for every  ,   is the monomial with   in the  th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of  .[1]: 371 

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

 [1]: 372 

Twisting

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

If   is a quasi-bialgebra and   is an invertible element such that  , set

 
 

Then, the set   is also a quasi-bialgebra obtained by twisting   by F, which is called a twist or gauge transformation.[1]: 373  If   was a braided quasi-bialgebra with universal R-matrix   , then so is   with universal R-matrix   (using the notation from the above section).[1]: 376  However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by   and then   is equivalent to twisting by  , and twisting by   then   recovers the original quasi-bialgebra.

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

Theorem: Let  ,   be quasi-bialgebras, let   be the twisting of   by  , and let there exist an isomorphism:  . Then the induced tensor functor   is a tensor category equivalence between   and  . Where  . Moreover, if   is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.[1]: 375–376 

Usage

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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

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References

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  1. ^ a b c d e f g h C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Springer-Verlag. ISBN 0387943706

Further reading

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  • 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