---
tag: 663BP8ZHS5
kind: concept
name: Yang–Baxter equation
msc: [16T25, 81R12, 82B23]
sources:
  - url: https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation
same_as:
  wikidata: Q4476530
  wikipedia: Yang–Baxter equation
---

## Summary

The Yang–Baxter equation is the identity $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ for a linear map $R$ of $V \otimes V$.
The indices say on which two of three tensor factors $R$ acts.
It first appeared in exactly solvable models of statistical mechanics and in quantum scattering in one dimension.
Composing with the flip turns a solution into a solution of the braid relation.
So an invertible solution gives representations of the [[braid group|BNR4H5GCTP]].
The universal R-matrix of a [[quasitriangular Hopf algebra|NCRPWS9W5R]] satisfies it, and [[quantum groups|AP2ZDXANKC]] are a rich source of solutions.

## Explanations

### L1 human

Picture three particles on a line that pass each other, and let $R$ describe how two of them interact when they cross.
The three particles can be brought from one order to the reverse order in two different sequences of pairwise crossings.
The Yang–Baxter equation says that both sequences give the same result.
In pictures of braids, it says that a strand can slide across the crossing of two other strands.
The plain swap of tensor factors is the simplest solution.