Yang–Baxter equation

Let ${\displaystyle A}$ be a unital associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for ${\displaystyle R(u,u')}$, a parameter-dependent element of the tensor product ${\displaystyle A\otimes A}$ (here, ${\displaystyle u}$ and ${\displaystyle u'}$ are the parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers ℝ⁺ in the case of a multiplicative parameter).

Let ${\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))}$ for ${\displaystyle i,j=1,...,3}$, with algebra homomorphisms ${\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A}$ determined by

${\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}$

${\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}$

${\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}$

The general form of the Yang–Baxter equation is

${\displaystyle R_{12}(u_{1},u_{2})\ R_{13}(u_{1},u_{3})\ R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})\ R_{13}(u_{1},u_{3})\ R_{12}(u_{1},u_{2}),}$

for all values of ${\displaystyle u_{1}}$ , ${\displaystyle u_{2}}$ and ${\displaystyle u_{3}}$ .

Let ${\displaystyle V}$ be a module of ${\displaystyle A}$, and ${\displaystyle P_{ij}=\phi _{ij}(P)}$ . Let ${\displaystyle P:V\otimes V\to V\otimes V}$ be the linear map satisfying ${\displaystyle P(x\otimes y)=y\otimes x}$ for all ${\displaystyle x,y\in V}$. The Yang–Baxter equation then has the following alternate form in terms of ${\displaystyle {\check {R}}(u,u')=P\circ R(u,u')}$ on ${\displaystyle V\otimes V}$.

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u_{1},u_{2}))({\check {R}}(u_{1},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{2},u_{3}))=({\check {R}}(u_{2},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{1},u_{3}))({\check {R}}(u_{1},u_{2})\otimes \mathbf {1} )}$.

Alternatively, we can express it in the same notation as above, defining ${\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))}$ , in which case the alternate form is

${\displaystyle {\check {R}}_{23}(u_{1},u_{2})\ {\check {R}}_{12}(u_{1},u_{3})\ {\check {R}}_{23}(u_{2},u_{3})={\check {R}}_{12}(u_{2},u_{3})\ {\check {R}}_{23}(u_{1},u_{3})\ {\check {R}}_{12}(u_{1},u_{2}).}$

In the parameter-independent special case where ${\displaystyle {\check {R}}}$ does not depend on parameters, the equation reduces to

${\displaystyle (\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})=({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )}$,

and a representation of the braid group, ${\displaystyle B_{n}}$, can be constructed on ${\displaystyle V^{\otimes n}}$ by ${\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}}$ for ${\displaystyle i=1,\dots ,n-1}$. This representation can be used to determine quasi-invariants of braids, knots and links.

A common ansatz for computing solutions is the difference property, ${\displaystyle R(u,u')=R(u-u')}$ , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization ${\displaystyle R(u,u')=R(u/u')}$ , in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations:

${\displaystyle R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),}$

for all values of ${\displaystyle u}$ and ${\displaystyle v}$. For a multiplicative parameter, the Yang–Baxter equation is

${\displaystyle R_{12}(u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),}$

for all values of ${\displaystyle u}$ and ${\displaystyle v}$.

The braided forms read as:

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(u+v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u+v))({\check {R}}(u)\otimes \mathbf {1} )}$

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(uv)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(uv))({\check {R}}(u)\otimes \mathbf {1} )}$

In some cases, the determinant of ${\displaystyle R(u)}$ can vanish at specific values of the spectral parameter ${\displaystyle u=u_{0}}$. Some ${\displaystyle R}$ matrices turn into a one dimensional projector at ${\displaystyle u=u_{0}}$. In this case a quantum determinant can be defined .

• Lie bialgebra
• Yangian
• Reidemeister move
• Quasitriangular Hopf algebra

• Jimbo, Michio (1989). "Introduction to the Yang-Baxter equation". International Journal of Modern Physics A. 4 (15): 3759–3777. doi:10.1142/S0217751X89001503. MR 1017340.
• H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989, Springer-Verlag Berlin, ISBN 3-540-53503-9.
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
• Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), arXiv:math-ph/0606053.

• Hazewinkel, Michiel, ed. (2001) [1994], "Yang-Baxter equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4