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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 ${\displaystyle k}$, Yang-Baxter operators are ${\displaystyle k}$-linear mappings ${\displaystyle R:V\otimes _{k}V\rightarrow V\otimes _{k}V}$. The operator ${\displaystyle R}$ satisfies the quantum Yang-Baxter equation if

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

where

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

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

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

See also

• Yang-Baxter equation
• Hopf algebra

References

• Geometric topology