Yang–Baxter operators are

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

endomorphisms In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a grou ...

with applications in

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

. They are named after

theoretical physicists The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this art ...

Yang Chen-Ning Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge th ...

and

Rodney Baxter Rodney James Baxter FRS FAA (born 8 February 1940 in London, United Kingdom) is an Australian physicist, specialising in statistical mechanics. He is well known for his work in exactly solved models, in particular vertex models such as the ...

. These

operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...

are particularly notable for providing solutions to the quantum

Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their ...

, which originated in

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, and for their use in constructing invariants of

knots A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...

, links, and three-dimensional

manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

.Baxter, R. (1982). "Exactly solved models in statistical mechanics". Academic Press. ISBN 978-0-12-083180-7.Yang, C.N. (1967). "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction". ''

Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. The journal is considered one of the most prestigious in the field of physics ...

''. 19: 1312–1315.Kauffman, L.H. (1991). "Knots and physics". Series on Knots and Everything. 1. World Scientific. ISBN 978-981-02-0332-1.

Definition

In the category of left modules over a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

k

, Yang–Baxter operators are

k

-linear mappings

R: V \otimes_k V \rightarrow V \otimes_k V

. The operator

R

satisfies the ''quantum Yang-Baxter equation'' if

$R_R_R_ = R_R_R_$

where

$R_ = R \otimes_k 1$ ,
$R_ = 1 \otimes_k R$ ,
$R_ = (1 \otimes_k \tau_)(R \otimes_k 1)(1 \otimes_k \tau_)$

The

\tau_

represents the "twist" mapping defined for

k

-modules

U

and

V

\tau_(u \otimes v) = v \otimes u

for all

u \in U

and

v \in V

. An important relationship exists between the quantum Yang-Baxter equation and the braid equation. If

R

satisfies the quantum Yang-Baxter equation, then

B = \tau_R

satisfies

B_B_B_ = B_B_B_

.Joyal, A.; Street, R. (1993). "Braided tensor categories". ''

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

''. 102: 20–78.

Applications

Yang–Baxter operators have applications in

and

.Zamolodchikov, A.B.; Zamolodchikov, A.B. (1975). "Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models". ''

Annals of Physics ''Annals of Physics'' is a monthly peer-reviewed scientific journal covering all aspects of physics. It was established in 1957 and is published by Elsevier. The editor-in-chief is Neil Turok ( University of Edinburgh School of Physics and Astrono ...

''. 120: 253–291.Jimbo, M. (1985). "A q-difference analogue of U(g) and the Yang-Baxter equation". ''

Letters in Mathematical Physics ''Letters in Mathematical Physics'' is a peer-reviewed scientific journal in mathematical physics published by Springer Science+Business Media. It publishes letters and longer research articles, occasionally also articles containing topical reviews ...

''. 10: 63–69.Reshetikhin, N.Yu.; Turaev, V.G. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups". ''

Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing ...

''. 103: 547–597.

References

{{reflist Geometric topology Morphisms Theoretical physics

Definition

Applications

See also

References