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In their most familiar form, Yang–Baxter operators are certain invertible linear endomorphisms which have applications to physics and topology. In physics these operators often provide solutions to the quantum [[Yang–Baxter equation|Yang–Baxter equation]], an equation which has its roots in statistical mechanics [[#References|[a30]]], [[#References|[a7]]], [[#References|[a28]]], [[#References|[a31]]], [[#References|[a35]]] (cf. also [[Statistical mechanics, mathematical problems in|Statistical mechanics, mathematical problems in]]). In topology quite often they can be used to construct invariants of knots, links or three-dimensional manifolds (cf. also [[Knot theory|Knot theory]]; [[Link|Link]]; [[Three-dimensional manifold|Three-dimensional manifold]]); cf. [[#References|[a30]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a26]]], [[#References|[a27]]], [[#References|[a32]]], [[#References|[a33]]].

Closely related to the quantum Yang–Baxter equation is the braid equation. There are natural categorical structures associated with the braid and quantum Yang–Baxter equations which play an important role in [[Quantum groups|quantum groups]] and their applications [[#References|[a21]]], [[#References|[a23]]], [[#References|[a5]]], [[#References|[a34]]].

Yang–Baxter operators in the [[Category|category]] $\square _ { k }\operatorname {Mod}$ of left modules over a [[Commutative ring|commutative ring]] $k$ are certain $k$-linear mappings $R : V \otimes _ { k } V \rightarrow V \otimes _ { k } V$. Let $R _ { 12 } = R \otimes _ { k } 1$, $R _ { 23 } = 1 \otimes _ { k } R$ and $R _ { 13 } = ( 1 \otimes _ { k } \tau _ { V , V } ) ( R \otimes _ { k } 1 ) ( 1 \otimes _ { k } \tau _ { V , V } )$, where $\tau _ { U , V } : U \otimes _ { k } V \rightarrow V \otimes _ { k } U$ is the  "twist"  mapping defined for $k$-modules $U$ and $V$ by $\tau_{ U , V } ( u \otimes v ) = v \otimes u$ for all $u \in U$ and $v \in V$. Then $R$ satisfies the quantum Yang–Baxter equation in $\square _ { k }\operatorname {Mod}$ if

\begin{equation} \tag{a1} R _ { 12 } R _ { 13 } R _ { 23 } = R _ { 23 } R _ { 13 } R _ { 12 }. \end{equation}

Note that $R$ satisfies (a1) if and only if $B = \tau _ { V , V }  R$ satisfies the braid equation in $\square _ { k }\operatorname {Mod}$, which is

\begin{equation} \tag{a2} B _ { 12 } B _ { 23 } B _ { 12 } = B _ { 23 } B _ { 12 } B _ { 23 }. \end{equation}

If $R$ is invertible and satisfies (a2), that is, $R _ { 12 } R _ { 23 } R _ { 12 } = R _ { 23 } R _ { 12 } R _ { 23 }$, then $R$ is a Yang–Baxter operator in $\square _ { k }\operatorname {Mod}$ (see [[#References|[a5]]]). There are other formulations of the notion of Yang–Baxter operator in the context of modules; see, e.g., [[#References|[a8]]] and [[#References|[a9]]].

Observe that the quantum Yang–Baxter and braid equations have natural formulations in any category $\mathcal{C}$ with a suitable notion of tensor product and in which the tensor product of morphisms is defined [[#References|[a16]]], [[#References|[a20]]], [[#References|[a34]]]. The notion of quantum Yang–Baxter operator thus has a natural generalization to categories $\mathcal{C}$ with such additional structure; see, e.g., [[#References|[a16]]], [[#References|[a5]]].

A good source of solutions to (a1) in $\square _ { k }\operatorname {Mod}$ are certain elements $R = \sum _ { s = 1 } ^ { n } a _ { s } \otimes b _ { s } \in A \otimes _ { k } A$, where $A$ is an [[Algebra|algebra]] over $k$. For $R \in A \otimes _ { k } A$ and a $k$-module $V$, let $R _ { V } : V \otimes _ { k } V \rightarrow V \otimes _ { k } V$ be defined by $R _ { V } ( u \otimes v ) = R \cdot ( u \otimes v )$ for all $u , v \in V$, where $V \otimes _ { k } V$ is regarded as a left $A \otimes _ { k } A$-module under component multiplication. Then $R _ { V }$ is a solution to (a1) for all left $k$-modules $V$ if and only if

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001044.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a3)</td></tr></table>

When $A = M _ { n } ( k )$ is the algebra of $( n \times n )$-matrices over $k$, then an $R \in A \otimes _ { k } A$ which satisfies (a1), or equivalently (a3), is called an $R$-matrix. Suppose that $A = M _ { n } ( k )$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001052.png"/>, where $r _ { t \text{l}} ^ { s k } \in  k $ and $\{ E _ { t } ^ { s } \} _ { 1  \leq s , t \leq n}$ is the standard basis for $A$. Then (a3) is equivalent to

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001056.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a4)</td></tr></table>

for all $1 \leq p , q , r , a , b , c \leq n$, which is probably the most familiar form of the quantum Yang–Baxter equation. Ordinarily, (a4) is written using the Einstein [[summation convention]], that is, summation signs are omitted with the understanding that indices that appear as upper and lower indices are summed over their full range of values.

Certain $k$-algebras $A$ with an $R \in A \otimes _ { k } A$ which satisfies (a1) can be used to construct invariants. Quasi-triangular Hopf algebras, in particular quantum algebras (cf. also [[Quasi-triangular Hopf algebra|Quasi-triangular Hopf algebra]]), give rise to regular isotopy invariants of $1$-$1$ tangles. Ribbon Hopf algebras give rise to regular isotopy invariants of knots and links, and under mild restrictions they give rise to invariants of three-dimensional manifolds. Let $k$ be a field. In this case a finite-dimensional [[Hopf algebra|Hopf algebra]] $A$ over $k$ is closely linked to these structures. The Hopf algebra $A$ is a subHopf algebra of the quantum double $D ( A )$ of $A$, which is a quasi-triangular Hopf algebra [[#References|[a1]]]. Every finite-dimensional quasi-triangular Hopf algebra over $k$ is a subHopf algebra of a ribbon Hopf algebra; in particular, $A$ is a subHopf algebra of a ribbon Hopf algebra [[#References|[a6]]].

The classification of $R$-matrices seems to be a very daunting task, and most work to date (1998) has involved symbolic computation. Suppose that $k$ is the field of complex numbers. Then the $R$-matrices are completely classified in the $n = 2$ case [[#References|[a13]]] and the classification of one basic family is known in the $n = 3$ case [[#References|[a14]]].

Some of the more important examples of $R$-matrices, those related to the quantized enveloping algebras, are formal infinite sums or belong to a completed tensor product. See [[#References|[a4]]], [[#References|[a1]]] for discussion of this important part of the theory.

There is a category $\square _ { A } ^ { A } \mathcal{C}$ with a pre-braiding structure, defined and studied in [[#References|[a34]]], associated to a bi-algebra $A$ over $k$ which gives rise to Yang–Baxter operators. Here, the formal variant $\square _ { A } \mathcal{C} ^ { A }$ is considered, whose objects are left $A$-modules and right $A$-comodules $V$ which satisfy the condition

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001084.png"/></td> </tr></table>

for all $a \in A$ and $v \in V$, where $\rho ( v ) = v ^ { ( 1 ) } \otimes  v ^ { ( 2 ) } \in V \otimes _ { k } A$ denotes the [[Coproduct|coproduct]] $\rho$ applied to $v \in V$. For an object $V$ of $\square _ { A } \mathcal{C} ^ { A }$, define $\mathcal{R} _ { V } : V \otimes _ { k } V \rightarrow V \otimes _ { k } V$ by $\mathcal{R} _ { V } ( u \otimes v ) = u ^ { \{ 1 \} } \otimes u ^ { ( 2 ) } . v$ for all $u , v \in V$. Then $\mathcal{R} _ { V }$ satisfies (a1). The pre-braiding structure on $\square _ { A } \mathcal{C} ^ { A }$ is the collection of morphisms of the form $\sigma _ { U , V } : U \otimes _ { k } V \rightarrow V \otimes _ { k } U$ which are defined for all pairs of objects $U$, $V$ by $\sigma_{ U , V}  ( u \otimes v ) = u ^ { ( 2 ) } . v \otimes u ^ { ( 1 ) }$ for all $u \in U$ and $v \in V$. Observe that $\sigma_{ V , V } = \tau_{ V , V} R _ { V }$ is a solution to the braid equation (a2). When $A$ is a [[Hopf algebra|Hopf algebra]], the morphism $\sigma_{  U, V}$ is invertible, and the collection of all $\sigma_{  U, V}$ is referred to as a braiding structure. When $k$ is a field and $A$ is a finite-dimensional Hopf algebra over $k$, the category $\square _ { D(A) } \mathcal{C}^{D ( A )}$ can be identified with $\square _{D(A)} \operatorname{Mod}$, the category of left modules of the quantum double [[#References|[a23]]].

The FRT construction of [[#References|[a11]]], [[#References|[a10]]] has an interesting interpretation in light of the preceding paragraph. Suppose that $k$ is a field and $R : V \otimes _ { k } V \rightarrow V \otimes _ { k } V$ is a solution to (a1), where $V$ is a finite-dimensional [[Vector space|vector space]] over $k$. The FRT construction $A ( R )$ is a certain bi-algebra over $k$ associated with $R$. There is a natural way of turning $V$ into an object of $\square _ { A ( R ) } {\cal C} ^ { A ( R) }$ such that $R = \mathcal{R} _ { \mathcal{V} }$, described in [[#References|[a24]]]. For a universal description of the FRT construction associated with certain Yang–Baxter operators, see [[#References|[a5]]], [[#References|[a21]]]. See also [[#References|[a8]]] for a discussion of algebras associated with Yang–Baxter operators.

There is a certain quotient $\widetilde { A  ( R )}$ of $A ( R )$ which is more closely tied to $R$ from a computational point of view. If $V \neq ( 0 )$, then it is never the case that $A ( R )$ is a Hopf algebra, whereas $\widetilde { A  ( R )}$ may very well be a Hopf algebra [[#References|[a25]]]. Determining new families of solutions to (a1) of the type described in the preceding paragraph may very well involve using a combination of bi-algebra techniques involving $\widetilde { A  ( R )}$ and computer methods [[#References|[a3]]], [[#References|[a25]]].

There are parametrized versions of (a1), and hence parametrized versions of Yang–Baxter operators. Let $X$ be a set, $R : X \rightarrow \operatorname { End } _ { k } ( V \otimes _ { k } V )$ be a function and suppose that $Z \subseteq X \times X$ is a non-empty subset with a (multiplication) mapping $Z \rightarrow X$. Then $R$ satisfies the $1$-parameter quantum Yang–Baxter equation if

\begin{equation*} R ( x ) _ { 12 } R ( x y ) _ { 13 } R ( y ) _ { 23 } = R ( y ) _ { 23 } R ( x y ) _ { 13 } R ( x ) _ { 12 } \end{equation*}

holds for all $( x , y ) \in Z$. There is an FRT construction for $1$-parameter families [[#References|[a3]]]. A $2$-parameter family of solutions to the quantum Yang–Baxter equation is a function $R : X \times X \rightarrow \operatorname { End } _ { k } ( V \otimes _ { k } V )$ which satisfies

\begin{equation*} R ( x , y ) _ { 12 } R ( x , z ) _ { 13 } R ( y , z ) _ { 23 } = \end{equation*}

\begin{equation*} = R ( y , z ) _ { 23 } R ( x , z ) _ { 13 } R ( x , y ) _ { 12 } \end{equation*}

for all $x , y , z \in X$. For examples and discussion, see [[#References|[a1]]], [[#References|[a12]]], [[#References|[a11]]], [[#References|[a15]]], [[#References|[a19]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  V.G. Drinfel'd,   "Quantum groups" , ''Proc. Internat. Congress Mathematicians Berkeley, California, (1987)'' , Amer. Math. Soc.  (1988)  pp. 798–820</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C. Kassel,   "Quantum groups" , ''Graduate Texts Math.'' , '''155''' , Springer  (1995)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L.A. Lambe,   D.E. Radford,   "Introduction to the quantum Yang–Baxter equation and quantum groups: An algebraic approach" , Kluwer Acad. Publ.  (1997)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V. Chari,   A. Pressley,   "A guide to quantum groups" , Cambridge Univ. Press  (1994)  (Corrected reprint: 1995)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Schauenburg,   "On coquasitriangular Hopf algebras and the quantum Yang–Baxter equation" , ''Algebra Berichte'' , '''67''' , R. Fischer  (1992)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  N.Yu. Reshetikhin,   V.G. Turaev,   "Ribbon graphs and their invariants derived from quantum groups"  ''Comm. Math. Phys.'' , '''127'''  (1990)  pp. 1–26</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  R. Baxter,   "Exactly solved models in stastistical mechanics" , Acad. Press  (1982)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  Yu.I. Manin,   "Quantum groups and noncommutative geometry"  ''Centre de Recherche Math. Univ. Montreal''  (1988)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  V.G. Turaev,   "The Yang–Baxter equation and invariants of links"  ''Invent. Math.'' , '''92''' :  3  (1988)  pp. 527–553</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  L.D. Faddeev,   N.Yu. Reshetikhin,   L.A. Takhtadzhan,   "Quantization of Lie groups and Lie algebras"  ''Leningrad Math. J.'' , '''1'''  (1990)  pp. 193–225  ''Algebra Anal.'' , '''1'''  (1989)  pp. 178–206</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  L.D. Faddeev,   N.Yu. Reshetikhin,   L.A. Takhtadzhan,   "Quantum groups" , ''Braid group, knot theory and statistical mechanics'' , ''Adv. Ser. Math. Phys.'' , '''9''' , World Sci.  (1989)  pp. 97–110</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  L.D. Faddeev,   N.Yu. Reshetikhin,   L.A. Takhtadzhan,   "Quantization of Lie groups and Lie algebras" , ''Algebraic analysis, Papers Dedicated to Prof. Mikio Sato on the Occasion of his Sixtieth Birthday'' , '''I''' , Acad. Press  (1988)  pp. 129–139</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  J. Hietarinta,   "Solving the two-dimensional constant quantum Yang-Baxter equation"  ''J. Math. Phys.'' , '''34'''  (1993)  pp. 1725–1756</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  J. Hietarinta,   "The upper triangular solutions to the three-state constant quantum Yang-Baxter equation"  ''J. Phys. A'' , '''26'''  (1993)  pp. 7077–7095</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  M. Jimbo,   "A $q$-difference analogue of $U ( g )$ and the Yang–Baxter equation"  ''Lett. Math. Phys.'' , '''10'''  (1985)  pp. 63–69</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  A. Joyal,   R. Street,   "Braided tensor categories"  ''Adv. Math.'' , '''102'''  (1993)  pp. 20–78</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  L.H. Kauffman,   "Knots and physics" , ''Ser. Knots and Everything'' , '''1''' , World Sci.  (1991)</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  L.H. Kauffman,   S. Lins,   "Temperley–Lieb recoupling theory and invariants of $3$-manifolds" , ''Ann. Math. Studies'' , '''134''' , Princeton Univ. Press  (1994)</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  P.P. Kulish,   N.Yu. Reshetikhin,   E.K. Sklyanin,   "Yang–Baxter equation and representation theory: I"  ''Lett. Math. Phys.'' , '''5'''  (1981)  pp. 393–403</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  L.A. Lambe,   D.E. Radford,   "Algebraic aspects of the quantum Yang-Baxter equation"  ''J. Algebra'' , '''154'''  (1993)  pp. 228–288</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  R.G. Larson,   J. Towber,   "Two dual classes of bialgebras related to the concepts of `quantum groups' and `quantum lie algebras'"  ''Commun. Algebra'' , '''19'''  (1991)  pp. 3295–3345</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  S. Majid,   "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  S. Majid,   "Doubles of quasi triangular Hopf algebras"  ''Commun. Algebra'' , '''19'''  (1991)  pp. 3061–3073</td></tr><tr><td valign="top">[a24]</td> <td valign="top">  D.E. Radford,   "Solutions to the quantum Yang–Baxter equation and the Drinfel'd double"  ''J. Algebra'' , '''161'''  (1993)  pp. 20–32</td></tr><tr><td valign="top">[a25]</td> <td valign="top">  D.E. Radford,   "Solutions to the quantum Yang–Baxter equation arising from pointed bi-algebras"  ''Trans. Amer. Math. Soc.'' , '''343'''  (1994)  pp. 455–477</td></tr><tr><td valign="top">[a26]</td> <td valign="top">  N. Reshetikhin,   "Invariants of links and $3$-manifolds related to quantum groups" , ''Proc. Internat. Congr. Math., Kyoto, August, 21-29, 1990'' , '''II''' , Springer  (1991)  pp. 1373–1375</td></tr><tr><td valign="top">[a27]</td> <td valign="top">  N.Yu. Reshetikhin,   V.G. Turaev,   "Invariants of $3$-manifolds via link polynomials and quantum groups"  ''Invent. Math.'' , '''103'''  (1991)  pp. 547–597</td></tr><tr><td valign="top">[a28]</td> <td valign="top">  G.M. D'Ariano,   A. Montorsi,   M.G. Rasetti,   "Integrable systems in statistical mechanics" , ''Ser. Adv. Statist. Mech.'' , '''1''' , World Sci.  (1985)</td></tr><tr><td valign="top">[a29]</td> <td valign="top">  S. Shnider,   S. Sternberg,   "Quantum groups. From coalgebras to Drinfel'd algebras. A guided tour" , ''Graduate Texts Math. Phys.'' , '''II''' , Internat. Press  (1993)</td></tr><tr><td valign="top">[a30]</td> <td valign="top">  Y. Akutsu,   T. Deguchi,   M. Wadati,   "Exactly solvable models and knot theory"  ''Physics Reports. A Review Section of Physics Lett.'' , '''180'''  (1989)  pp. 247–332</td></tr><tr><td valign="top">[a31]</td> <td valign="top">  C.N. Yang,   "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction"  ''Phys. Rev. Lett.'' , '''19'''  (1967)  pp. 1312–1315</td></tr><tr><td valign="top">[a32]</td> <td valign="top">  C.N. Yang,   M.L. Ge,   "Braid group, knot theory and statistical mechanics" , ''Adv. Ser. Math. Phys.'' , '''9''' , World Sci.  (1989)</td></tr><tr><td valign="top">[a33]</td> <td valign="top">  C.N. Yang,   M.L. Ge,   "Braid group, knot theory and statistical mechanics, II" , ''Adv. Ser. Math. Phys.'' , '''17''' , World Sci.  (1994)</td></tr><tr><td valign="top">[a34]</td> <td valign="top">  D.N. Yetter,   "Quantum groups and representations of monoidal categories"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 261–290</td></tr><tr><td valign="top">[a35]</td> <td valign="top">  A.B. Zamolodchikov,   A.B. Zamolodchikov,   "Factorized $S$-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models"  ''Ann. Phys.'' , '''120'''  (1975)  pp. 253–291</td></tr></table>