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, an equation which has its roots in statistical mechanics [a30], [a7], [a28], [a31], [a35] (cf. also 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; Link; Three-dimensional manifold); cf. [a30], [a17], [a18], [a26], [a27], [a32], [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 and their applications [a21], [a23], [a5], [a34].

Yang–Baxter operators in the category of left modules over a commutative ring are certain -linear mappings . Let , and , where is the  "twist"  mapping defined for -modules and by for all and . Then satisfies the quantum Yang–Baxter equation in if

(a1)

Note that satisfies (a1) if and only if satisfies the braid equation in , which is

(a2)

If is invertible and satisfies (a2), that is, , then is a Yang–Baxter operator in (see [a5]). There are other formulations of the notion of Yang–Baxter operator in the context of modules; see, e.g., [a8] and [a9].

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

A good source of solutions to (a1) in are certain elements , where is an algebra over . For and a -module , let be defined by for all , where is regarded as a left -module under component multiplication. Then is a solution to (a1) for all left -modules if and only if

(a3)

When is the algebra of -matrices over , then an which satisfies (a1), or equivalently (a3), is called an -matrix. Suppose that and , where and is the standard basis for . Then (a3) is equivalent to

(a4)

for all , 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), 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 -algebras with an which satisfies (a1) can be used to construct invariants. Quasi-triangular Hopf algebras, in particular quantum algebras (cf. also Quasi-triangular Hopf algebra), give rise to regular isotopy invariants of - 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 be a field. In this case a finite-dimensional Hopf algebra over is closely linked to these structures. The Hopf algebra is a subHopf algebra of the quantum double of , which is a quasi-triangular Hopf algebra [a1]. Every finite-dimensional quasi-triangular Hopf algebra over is a subHopf algebra of a ribbon Hopf algebra; in particular, is a subHopf algebra of a ribbon Hopf algebra [a6].

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

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

There is a category with a pre-braiding structure, defined and studied in [a34], associated to a bi-algebra over which gives rise to Yang–Baxter operators. Here, the formal variant is considered, whose objects are left -modules and right -comodules which satisfy the condition for all and , where denotes the coproduct applied to . For an object of , define by for all . Then satisfies (a1). The pre-braiding structure on is the collection of morphisms of the form which are defined for all pairs of objects , by for all and . Observe that is a solution to the braid equation (a2). When is a Hopf algebra, the morphism is invertible, and the collection of all is referred to as a braiding structure. When is a field and is a finite-dimensional Hopf algebra over , the category can be identified with , the category of left modules of the quantum double [a23].

The FRT construction of [a11], [a10] has an interesting interpretation in light of the preceding paragraph. Suppose that is a field and is a solution to (a1), where is a finite-dimensional vector space over . The FRT construction is a certain bi-algebra over associated with . There is a natural way of turning into an object of such that , described in [a24]. For a universal description of the FRT construction associated with certain Yang–Baxter operators, see [a5], [a21]. See also [a8] for a discussion of algebras associated with Yang–Baxter operators.

There is a certain quotient of which is more closely tied to from a computational point of view. If , then it is never the case that is a Hopf algebra, whereas may very well be a Hopf algebra [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 and computer methods [a3], [a25].

There are parametrized versions of (a1), and hence parametrized versions of Yang–Baxter operators. Let be a set, be a function and suppose that is a non-empty subset with a (multiplication) mapping . Then satisfies the -parameter quantum Yang–Baxter equation if holds for all . There is an FRT construction for -parameter families [a3]. A -parameter family of solutions to the quantum Yang–Baxter equation is a function which satisfies for all . For examples and discussion, see [a1], [a12], [a11], [a15], [a19].

References

[a1]	V.G. Drinfel'd,   "Quantum groups" , Proc. Internat. Congress Mathematicians Berkeley, California, (1987) , Amer. Math. Soc.  (1988)  pp. 798–820
[a2]	C. Kassel,   "Quantum groups" , Graduate Texts Math. , 155 , Springer  (1995)
[a3]	L.A. Lambe,   D.E. Radford,   "Introduction to the quantum Yang–Baxter equation and quantum groups: An algebraic approach" , Kluwer Acad. Publ.  (1997)
[a4]	V. Chari,   A. Pressley,   "A guide to quantum groups" , Cambridge Univ. Press  (1994)  (Corrected reprint: 1995)
[a5]	P. Schauenburg,   "On coquasitriangular Hopf algebras and the quantum Yang–Baxter equation" , Algebra Berichte , 67 , R. Fischer  (1992)
[a6]	N.Yu. Reshetikhin,   V.G. Turaev,   "Ribbon graphs and their invariants derived from quantum groups"  Comm. Math. Phys. , 127  (1990)  pp. 1–26
[a7]	R. Baxter,   "Exactly solved models in stastistical mechanics" , Acad. Press  (1982)
[a8]	Yu.I. Manin,   "Quantum groups and noncommutative geometry"  Centre de Recherche Math. Univ. Montreal  (1988)
[a9]	V.G. Turaev,   "The Yang–Baxter equation and invariants of links"  Invent. Math. , 92 : 3  (1988)  pp. 527–553
[a10]	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
[a11]	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
[a12]	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
[a13]	J. Hietarinta,   "Solving the two-dimensional constant quantum Yang-Baxter equation"  J. Math. Phys. , 34  (1993)  pp. 1725–1756
[a14]	J. Hietarinta,   "The upper triangular solutions to the three-state constant quantum Yang-Baxter equation"  J. Phys. A , 26  (1993)  pp. 7077–7095
[a15]	M. Jimbo,   "A -difference analogue of and the Yang–Baxter equation"  Lett. Math. Phys. , 10  (1985)  pp. 63–69
[a16]	A. Joyal,   R. Street,   "Braided tensor categories"  Adv. Math. , 102  (1993)  pp. 20–78
[a17]	L.H. Kauffman,   "Knots and physics" , Ser. Knots and Everything , 1 , World Sci.  (1991)
[a18]	L.H. Kauffman,   S. Lins,   "Temperley–Lieb recoupling theory and invariants of -manifolds" , Ann. Math. Studies , 134 , Princeton Univ. Press  (1994)
[a19]	P.P. Kulish,   N.Yu. Reshetikhin,   E.K. Sklyanin,   "Yang–Baxter equation and representation theory: I"  Lett. Math. Phys. , 5  (1981)  pp. 393–403
[a20]	L.A. Lambe,   D.E. Radford,   "Algebraic aspects of the quantum Yang-Baxter equation"  J. Algebra , 154  (1993)  pp. 228–288
[a21]	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
[a22]	S. Majid,   "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)
[a23]	S. Majid,   "Doubles of quasi triangular Hopf algebras"  Commun. Algebra , 19  (1991)  pp. 3061–3073
[a24]	D.E. Radford,   "Solutions to the quantum Yang–Baxter equation and the Drinfel'd double"  J. Algebra , 161  (1993)  pp. 20–32
[a25]	D.E. Radford,   "Solutions to the quantum Yang–Baxter equation arising from pointed bi-algebras"  Trans. Amer. Math. Soc. , 343  (1994)  pp. 455–477
[a26]	N. Reshetikhin,   "Invariants of links and -manifolds related to quantum groups" , Proc. Internat. Congr. Math., Kyoto, August, 21-29, 1990 , II , Springer  (1991)  pp. 1373–1375
[a27]	N.Yu. Reshetikhin,   V.G. Turaev,   "Invariants of -manifolds via link polynomials and quantum groups"  Invent. Math. , 103  (1991)  pp. 547–597
[a28]	G.M. D'Ariano,   A. Montorsi,   M.G. Rasetti,   "Integrable systems in statistical mechanics" , Ser. Adv. Statist. Mech. , 1 , World Sci.  (1985)
[a29]	S. Shnider,   S. Sternberg,   "Quantum groups. From coalgebras to Drinfel'd algebras. A guided tour" , Graduate Texts Math. Phys. , II , Internat. Press  (1993)
[a30]	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
[a31]	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
[a32]	C.N. Yang,   M.L. Ge,   "Braid group, knot theory and statistical mechanics" , Adv. Ser. Math. Phys. , 9 , World Sci.  (1989)
[a33]	C.N. Yang,   M.L. Ge,   "Braid group, knot theory and statistical mechanics, II" , Adv. Ser. Math. Phys. , 17 , World Sci.  (1994)
[a34]	D.N. Yetter,   "Quantum groups and representations of monoidal categories"  Math. Proc. Cambridge Philos. Soc. , 108  (1990)  pp. 261–290
[a35]	A.B. Zamolodchikov,   A.B. Zamolodchikov,   "Factorized -matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models"  Ann. Phys. , 120  (1975)  pp. 253–291