Yang-Baxter operator

Yang-Baxter operators are invertible linear endomorphisms with applications in theoretical physics and topology named after theoretical physicists Yang Chen-Ning and Rodney Baxter. 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 $k$, 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

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

where

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

The $\tau _{U,V}$ represents the "twist" mapping defined for $k$-modules $U$ and $V$ by $\tau _{U,V}(u\otimes v)=v\otimes u$ 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 $B=\tau _{V,V}R$ satisfies $B_{12}B_{23}B_{12}=B_{23}B_{12}B_{23}$.

Applications

Yang-Baxter operators have applications in statistical mechanics and topology.

See also

• Yang-Baxter equation
• Hopf algebra
• Lie bialgebra
• Yangian
• Braid theory
• Quantum groups