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== Definition == == Definition ==
In the ] over a ] <math>k</math>, Yang-Baxter operators are <math>k</math>-linear mappings <math>R: V \otimes_k V \rightarrow V \otimes_k V</math>. The operator <math>R</math> satisfies the ''quantum Yang-Baxter equation'' if In the ] over a ] <math>k</math>, Yang-Baxter operators are ] <math>R: V \otimes_k V \rightarrow V \otimes_k V</math>. The operator <math>R</math> satisfies the ''quantum Yang-Baxter equation'' if
<blockquote><math>R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}</math></blockquote> <blockquote><math>R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}</math></blockquote>
where where

Revision as of 22:59, 28 December 2024

A mathematical operator used in theoretical physics and topology

Yang-Baxter operators are invertible linear endomorphisms with applications in theoretical physics and topology. These operators are particularly notable for providing solutions to the quantum Yang-Baxter equation, which originated in statistical mechanics, and for their use in constructing invariants of knots, links, and three-dimensional manifolds.

Definition

In the category of left modules over a commutative ring k {\displaystyle k} , Yang-Baxter operators are k {\displaystyle k} -linear mappings R : V ⊗ k V → V ⊗ k V {\displaystyle R:V\otimes _{k}V\rightarrow V\otimes _{k}V} . The operator R {\displaystyle R} satisfies the quantum Yang-Baxter equation if

R 12 R 13 R 23 = R 23 R 13 R 12 {\displaystyle R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}}

where

R 12 = R ⊗ k 1 {\displaystyle R_{12}=R\otimes _{k}1} ,
R 23 = 1 ⊗ k R {\displaystyle R_{23}=1\otimes _{k}R} ,
R 13 = ( 1 ⊗ k τ V , V ) ( R ⊗ k 1 ) ( 1 ⊗ k τ V , V ) {\displaystyle R_{13}=(1\otimes _{k}\tau _{V,V})(R\otimes _{k}1)(1\otimes _{k}\tau _{V,V})}

The τ U , V {\displaystyle \tau _{U,V}} represents the "twist" mapping defined for k {\displaystyle k} -modules U {\displaystyle U} and V {\displaystyle V} by τ U , V ( u ⊗ v ) = v ⊗ u {\displaystyle \tau _{U,V}(u\otimes v)=v\otimes u} for all u ∈ U {\displaystyle u\in U} and v ∈ V {\displaystyle v\in V} .

An important relationship exists between the quantum Yang-Baxter equation and the braid equation. If R {\displaystyle R} satisfies the quantum Yang-Baxter equation, then B = τ V , V R {\displaystyle B=\tau _{V,V}R} satisfies B 12 B 23 B 12 = B 23 B 12 B 23 {\displaystyle B_{12}B_{23}B_{12}=B_{23}B_{12}B_{23}} .

See also

References

  1. Baxter, R. (1982). "Exactly solved models in statistical mechanics". Academic Press. ISBN 978-0-12-083180-7.
  2. Yang, C.N. (1967). "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction". Physical Review Letters. 19: 1312–1315.
  3. Kauffman, L.H. (1991). "Knots and physics". Series on Knots and Everything. 1. World Scientific. ISBN 978-981-02-0332-1.
  4. Joyal, A.; Street, R. (1993). "Braided tensor categories". Advances in Mathematics. 102: 20–78.
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