View source for Yang-Baxter operators

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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200101.png" /> of left modules over a [[Commutative ring|commutative ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200102.png" /> are certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200103.png" />-linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200104.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200108.png" /> is the  "twist"  mapping defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y1200109.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001011.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001014.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001015.png" /> satisfies the quantum Yang–Baxter equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001016.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001018.png" /> satisfies (a1) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001019.png" /> satisfies the braid equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001020.png" />, which is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001022.png" /> is invertible and satisfies (a2), that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001024.png" /> is a Yang–Baxter operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001026.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001028.png" /> with such additional structure; see, e.g., [[#References|[a16]]], [[#References|[a5]]].

A good source of solutions to (a1) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001029.png" /> are certain elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001031.png" /> is an [[Algebra|algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001032.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001033.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001034.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001035.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001036.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001039.png" /> is regarded as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001040.png" />-module under component multiplication. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001041.png" /> is a solution to (a1) for all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001042.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001043.png" /> if and only if

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

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001045.png" /> is the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001046.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001047.png" />, then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001048.png" /> which satisfies (a1), or equivalently (a3), is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001050.png" />-matrix. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001054.png" /> is the standard basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001055.png" />. Then (a3) is equivalent to

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

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001057.png" />, which is probably the most familiar form of the quantum Yang–Baxter equation. Ordinarily, (a4) is written using the Einstein summation convention (cf. also [[Einstein rule|Einstein rule]]), 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001058.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001059.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001060.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001061.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001063.png" /> be a field. In this case a finite-dimensional [[Hopf algebra|Hopf algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001064.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001065.png" /> is closely linked to these structures. The Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001066.png" /> is a subHopf algebra of the quantum double <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001067.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001068.png" />, which is a quasi-triangular Hopf algebra [[#References|[a1]]]. Every finite-dimensional quasi-triangular Hopf algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001069.png" /> is a subHopf algebra of a ribbon Hopf algebra; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001070.png" /> is a subHopf algebra of a ribbon Hopf algebra [[#References|[a6]]].

The classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001071.png" />-matrices seems to be a very daunting task, and most work to date (1998) has involved symbolic computation. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001072.png" /> is the field of complex numbers. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001073.png" />-matrices are completely classified in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001074.png" /> case [[#References|[a13]]] and the classification of one basic family is known in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001075.png" /> case [[#References|[a14]]].

Some of the more important examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001076.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001077.png" /> with a pre-braiding structure, defined and studied in [[#References|[a34]]], associated to a bi-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001078.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001079.png" /> which gives rise to Yang–Baxter operators. Here, the formal variant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001080.png" /> is considered, whose objects are left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001081.png" />-modules and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001082.png" />-comodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001083.png" /> which satisfy the condition

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

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001087.png" /> denotes the [[Coproduct|coproduct]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001088.png" /> applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001089.png" />. For an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001091.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001092.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001093.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001094.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001095.png" /> satisfies (a1). The pre-braiding structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001096.png" /> is the collection of morphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001097.png" /> which are defined for all pairs of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y12001099.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010100.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010102.png" />. Observe that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010103.png" /> is a solution to the braid equation (a2). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010104.png" /> is a [[Hopf algebra|Hopf algebra]], the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010105.png" /> is invertible, and the collection of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010106.png" /> is referred to as a braiding structure. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010107.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010108.png" /> is a finite-dimensional Hopf algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010109.png" />, the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010110.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010111.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010112.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010113.png" /> is a solution to (a1), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010114.png" /> is a finite-dimensional [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010115.png" />. The FRT construction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010116.png" /> is a certain bi-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010117.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010118.png" />. There is a natural way of turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010119.png" /> into an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010120.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010121.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010123.png" /> which is more closely tied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010124.png" /> from a computational point of view. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010125.png" />, then it is never the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010126.png" /> is a Hopf algebra, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010127.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010128.png" /> and computer methods [[#References|[a3]]], [[#References|[a25]]].

There are parametrized versions of (a1), and hence parametrized versions of Yang–Baxter operators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010129.png" /> be a set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010130.png" /> be a function and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010131.png" /> is a non-empty subset with a (multiplication) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010132.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010133.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010134.png" />-parameter quantum Yang–Baxter equation if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010135.png" /></td> </tr></table>

holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010136.png" />. There is an FRT construction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010137.png" />-parameter families [[#References|[a3]]]. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010138.png" />-parameter family of solutions to the quantum Yang–Baxter equation is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010139.png" /> which satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010140.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010141.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120010/y120010142.png" />. For examples and discussion, see [[#References|[a1]]], [[#References|[a12]]], [[#References|[a11]]], [[#References|[a15]]], [[#References|[a19]]].

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