4
$\begingroup$

Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities:

  1. $(X,\circ,1)$ is a monoid,

  2. $f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$

  3. $x\circ y=f(x,y)\circ g(x,y)$

  4. $g(x,y\circ z)=g(g(x,y),z)$

  5. $f(x\circ y,z)=f(x,f(y,z))$

  6. $f(x,y\circ z)=f(x,y)\circ f(g(x,y),z)$

  7. $g(x\circ y,z)=g(x,f(y,z))\circ g(y,z)$,

  8. $g(f(x,y),f(g(x,y),z))=f(g(x,f(y,z)),g(y,z))$.

Suppose that $T:X^{2}\rightarrow X^{2}$ is a function. Then we say that $T$ satisfies the Yang-Baxter equations if $(T\times 1_{X})\circ(1_{X}\times T)\circ(T\times 1_{X})=(1_{X}\times T)\circ(T\times 1_{X})\circ(1_{X}\times T)$.

If $(X,f,g,\circ,1)$ is a Yang-Baxter monoid and $T:X^{2}\rightarrow X^{2},T(x,y)=(f(x,y),g(x,y))$, then $T$ automatically satisfies the Yang-Baxter equations.

An LD-monoid is an algebra $(X,*,\circ,1)$ that satisfies the identities

  1. $(X,\circ,1)$ is a monoid,

  2. $x\circ y=(x*y)\circ x$,

  3. $x*(y\circ z)=(x*y)\circ (x*z)$,

  4. $x*(y*z)=(x\circ y)*z$,

  5. $x*(y*z)=(x*y)*(x*z)$.

  6. $x*1=1,1*x=x$.

For example, let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$ and let $*$ be the operation on $\mathcal{E}_{\lambda}$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Then $(\mathcal{E}_{\lambda},*,\circ,1)$ is an LD-monoid. If $(G,\circ,1)$ is a group and $x*y=xyx^{-1}$, then $(G,*,\circ,1)$ is always an LD-monoid. Suppose that $(X,\circ,1)$ is a monoid, $*$ is a binary operation, and $f(x,y)=x*y,g(x,y)=x$. If $f(x,y)=x*y,g(x,y)=x$, then $(X,*,\circ,1)$ is an LD-monoid precisely when $(X,f,g,\circ,1)$ is a Yang-Baxter monoid.

The motivation for identities 1-8 comes from the notion of a permutative Yang-Baxter monoid which satisfies these identities.

What are some examples of Yang-Baxter monoids that do not trivially arise from LD-monoids? Are there any references for the notion of a Yang-Baxter monoid anywhere?

The closest thing that I found to the notion of a Yang-Baxter monoid is known as the structure group of a solution to the Yang-Baxter equation. The structure group of a function $T:X^{2}\rightarrow X^{2}$ that satisfies the Yang-Baxter equation is the group with a presentation consisting of the relations $xy=uv$ whenever $T(x,y)=(u,v)$ which is a version of property 3.

$\endgroup$
  • $\begingroup$ For me property 1 says that $(X,\circ,1)$ is an algebra (I guess you don't necessarily want $X$ to be an $R$-module for some commutative ring $R$). Are $f$ and $g$ in the first sentence also part of the data of the algebra? Should I think of them as alternative products with the same unit 1 as $\circ$, and if so, are they associative too? $\endgroup$ – Jules Lamers Jan 8 at 4:57
  • 1
    $\begingroup$ The word algebra is used in the universal algebraic sense and not the ring theoretic sense. By algebra, I only mean a set and a bunch of operations on that set. In the LD-monoid example, the operation $*$ is not associative nor commutative, so $f$ and $g$ do not have to satisfy any sort of associativity. By the axiom $x\circ y=f(x,y)\circ g(x,y)$, one should think of $f(x,y),g(x,y)$ as a sort of distinguished factorization of $x\circ y$. $\endgroup$ – Joseph Van Name Jan 8 at 21:04

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.