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In the paper The Amplituhedron , Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes). Let $Z$ be a $(k+m)\times …

I am trying to understand the root systems $E_{n}$, $n \ge 10$. In particular, I would like to find some references which describe the number of real roots and imaginary roots of a given degree. Cons …

Let \begin{align} Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2}. \end{align} This polylogarithm satisfies the following Abel identity: \begin{align} & Li_2(-x) + \log x \log y \\ & + Li_2(-y) + \lo …

I am reading the lecture notes INTRODUCTION TO DONALDSON–THOMAS INVARIANTS. I have a question in the end of page 1 about the proof of a map is an automorphism. Let $m>0$ be an integer. Let $\overline …

Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, Gr …

The classical Yang-Baxter equation is \begin{align} [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1) \end{align} What are the differences between this equation in the case of Lie …

In this paper, page 149, the super Jacobi identity is given by \begin{align} J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0. \end{align} But in this article, …

I have a question about the equation (1.24) in the paper about classical r-matrices. It is said that when we put $\overline{r} = Pr$ in the equation (1.24): $$ \overline{r}_{23}\overline{r}_{12}P_{23 …

Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.

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 cl …

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties: (1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$. (2) $[r_{12}, r_{ …

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum …