Quasi-bialgebra

Template:Short description 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 $\mathrm{\Phi }$ 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 ${{B}}_{{𝒜}}=({𝒜},\mathrm{\Delta },\epsilon ,\mathrm{\Phi },l,r)$ is an algebra ${𝒜}$ over a field $\mathbb{𝔽}$ equipped with morphisms of algebras

$\mathrm{\Delta }:{𝒜}\to {𝒜}{\otimes }{𝒜}$

$\epsilon :{𝒜}\to \mathbb{𝔽}$

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

$(id\otimes \mathrm{\Delta })\circ \mathrm{\Delta }(a)=\mathrm{\Phi }[(\mathrm{\Delta }\otimes id)\circ \mathrm{\Delta }(a)]{\mathrm{\Phi }}^{-1},\mathrm{\forall }a\in {𝒜}$

$[(id\otimes id\otimes \mathrm{\Delta })(\mathrm{\Phi })][(\mathrm{\Delta }\otimes id\otimes id)(\mathrm{\Phi })]=(1\otimes \mathrm{\Phi })[(id\otimes \mathrm{\Delta }\otimes id)(\mathrm{\Phi })](\mathrm{\Phi }\otimes 1)$

$(\epsilon \otimes id)(\mathrm{\Delta }a)=l^{-1}al,(id\otimes \epsilon )\circ \mathrm{\Delta }=r^{-1}ar,\mathrm{\forall }a\in {𝒜}$

$(id\otimes \epsilon \otimes id)(\mathrm{\Phi })=r\otimes l^{-1}.$

Where $\mathrm{\Delta }$ and $ϵ$ are called the comultiplication and counit, $r$ and $l$ are called the right and left unit constraints (resp.), and $\mathrm{\Phi }$ is sometimes called the Drinfeld associator.Template:Rp This definition is constructed so that the category ${𝒜}-Mod$ 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.Template:Rp 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.Template:Rp Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: $l=r=1$ and $\mathrm{\Phi }=1\otimes 1\otimes 1$.

Braided quasi-bialgebras

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

$({\mathrm{\Delta }}^{op})(a)=R\mathrm{\Delta }(a)R^{-1}$

$(id\otimes \mathrm{\Delta })(R)=({\mathrm{\Phi }}_{231}{)}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}({\mathrm{\Phi }}_{213}{)}^{-1}$

$(\mathrm{\Delta }\otimes id)(R)=({\mathrm{\Phi }}_{321})R_{13}({\mathrm{\Phi }}_{213}{)}^{-1}{R}_{23}{\mathrm{\Phi }}_{123}$

Where, for every $a_1\otimes ...\otimes a_k\in {{𝒜}}^{\otimes k}$, $a_{i_1{i}_2...i_n}$ 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 ${{𝒜}}^{\otimes k}$.Template:Rp

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

$R_{12}{\mathrm{\Phi }}_{321}{R}_{13}({\mathrm{\Phi }}_{132}{)}^{-1}{R}_{23}{\mathrm{\Phi }}_{123}={\mathrm{\Phi }}_{321}{R}_{23}({\mathrm{\Phi }}_{231}{)}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}$Template:Rp

Twisting

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

If ${{B}}_{{𝒜}}$ is a quasi-bialgebra and $F\in {𝒜}{\otimes }{𝒜}$ is an invertible element such that $(\epsilon \otimes id)F=(id\otimes \epsilon )F=1$, set

${\mathrm{\Delta }}^{\prime }(a)=F\mathrm{\Delta }(a)F^{-1},\mathrm{\forall }a\in {𝒜}$

${\mathrm{\Phi }}^{\prime }=(1\otimes F)((id\otimes \mathrm{\Delta })F)\mathrm{\Phi }((\mathrm{\Delta }\otimes id)F^{-1})(F^{-1}\otimes 1).$

Then, the set $({𝒜},{\mathrm{\Delta }}^{\prime },\epsilon ,{\mathrm{\Phi }}^{\prime })$ is also a quasi-bialgebra obtained by twisting ${{B}}_{{𝒜}}$ by F, which is called a twist or gauge transformation.Template:Rp If $({𝒜},\mathrm{\Delta },\epsilon ,\mathrm{\Phi })$ was a braided quasi-bialgebra with universal R-matrix $R$ , then so is $({𝒜},{\mathrm{\Delta }}^{\prime },\epsilon ,{\mathrm{\Phi }}^{\prime })$ with universal R-matrix $F_{21}R{F}^{-1}$ (using the notation from the above section).Template:Rp 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 $F_2{F}_1$, 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 ${{B}}_{{𝒜}}$, ${{B}}_{{{𝒜}}^{\prime }}$ be quasi-bialgebras, let ${ℬ}{{\text{'}}}_{{{𝒜}}^{\prime }}$ be the twisting of ${{B}}_{{{𝒜}}^{\prime }}$ by $F$, and let there exist an isomorphism: $\alpha :{{B}}_{{𝒜}}\to {ℬ}{{\text{'}}}_{{{𝒜}}^{\prime }}$. Then the induced tensor functor $({\alpha }^{*},id,{\varphi }_2^F)$ is a tensor category equivalence between ${{𝒜}}^{\prime }-mod$ and ${𝒜}-mod$. Where ${\varphi }_2^F(v\otimes w)=F^{-1}(v\otimes w)$. Moreover, if $\alpha$ is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.Template:Rp

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

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