Questions tagged [yang-baxter-equations]

In the classical equation, one looks for $R\in\Lambda^2\mathfrak g$ such that $$[R,R]=0,$$ where the bracket is Schouten's bracker in $\Lambda^\bullet\mathfrak g$, the exterior algebra on a Lie algebra $\mathfrak g$. In the quantum one (in its non-parametric form...), one looks for endomorphisms $R:V\otimes V\to V\otimes V$ of tensor squares of vector spaces $V$ such that $$R_{12} \ R_{13} \ R_{23} = R_{23} \ R_{13} \ R_{12},$$

Filter by
Sorted by
Tagged with
8 votes
0 answers
216 views

$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
Pulcinella's user avatar
  • 5,506
1 vote
0 answers
69 views

Problem in understanding Theorem $6.2.9$ from Chari and Pressley

The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
Anil Bagchi.'s user avatar
6 votes
2 answers
249 views

Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)

I'm interested in solutions to the Yang-Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
Lagrenge's user avatar
  • 423
5 votes
1 answer
688 views

What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?

For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if $$(R\otimes id)(id\otimes R)(R\otimes id) = (id\otimes R)(R\otimes ...
Jake Wetlock's user avatar
  • 1,114
7 votes
0 answers
356 views

When do two knots have isomorphic fundamental bikeis?

A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...
Keshav Srinivasan's user avatar
7 votes
1 answer
140 views

Set-theoretic solutions of YBE for $n=3$

Is there a list of all set theoretic solutions $S:X \times X \to X \times X$ of the YBE for $X=\{1,2,3\}$? Or is it known how many solutions there are? I mean, $S_9$ is big but maybe not too big to ...
Bipolar Minds's user avatar
5 votes
0 answers
75 views

Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
darij grinberg's user avatar
2 votes
0 answers
107 views

Is the action of free self-distributive algebras on racks computable in polynomial time?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the mapping where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Then $B_{\...
Joseph Van Name's user avatar
2 votes
0 answers
102 views

Does shifted conjugacy still give you free self-distributive algebras on one generator for quotient groups of the braid groups?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the group homomorphism where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ for all $i>0$. ...
Joseph Van Name's user avatar
5 votes
1 answer
243 views

Examples of Yang-Baxter monoids

Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities: $(X,\circ,1)$ is a monoid, $f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$ $x\circ y=f(x,y)\circ ...
Joseph Van Name's user avatar
6 votes
0 answers
105 views

Permutative Yang-Baxter monoids

Suppose that $f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$ are mappings such that $T(x,y)=(f(x,y),g(x,y))$. An element $1\in X$ is said to be an identity if $T(1,x)=(x,1),T(x,1)=(1,x)$. The ...
Joseph Van Name's user avatar
3 votes
1 answer
307 views

Yang-Baxter equation for the asymmetric simple exclusion process (ASEP)

On page 14 of Craig Tracy's slides on ASEP, it states that the $n$-particle boundary condition can be reduced to the 2-particle boundary condition due to the fact that the $S$-matrix satisfies the ...
Y. Pei's user avatar
  • 247
6 votes
1 answer
245 views

How can I verify that a given solution of the Quantum Yang-Baxter equation is associated to a given Lie algebra?

Take, for instance, the $R$ matrix, \begin{equation} R(u)=\begin{pmatrix}u+1 & 0 & 0 & 0\\0 & u & 1 & 0\\0 & 1 & u & 0\\0 & 0 & 0 & u+1\end{pmatrix}, \...
Ricardo Vieira's user avatar
4 votes
1 answer
134 views

Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra

I have a solution (a $R$ matrix) of the Yang-Baxter equation, \begin{equation} R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1}) \end{equation} that probably ...
Ricardo Vieira's user avatar
3 votes
0 answers
72 views

Partially permutative matrices

Let $V$ be a finite dimensional vector space over a field $K$. Then a map $L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...
Joseph Van Name's user avatar
2 votes
1 answer
228 views

How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...
Jianrong Li's user avatar
  • 6,101