Related papers: Notes on the quantum tetrahedron
We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes,…
We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a…
The eccentricity matrix $E(G)$ of a simple connected graph $G$ is obtained from the distance matrix $D(G)$ of $G$ by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses…
In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing…
We study quantum automorphism group of vertex-transitive graphs using intertwinner spaces of the magic unitary matrix associated to this quantum subgroups of $S_n^+$. We also give some applications to quantum symmetries of circulant graphs…
We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra,…
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra…
Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe…
We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group E6, and of its subgroups. We are therefore led…
Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_R$ of $R$ is a graph in which the vertex set comprises of set of all nonzero proper ideals of $R$ and two vertices $I$ and $K$ are adjacent if and only if…
We present some graphical characterizations of positive definite symmetric quasi-Cartan matrices of Dynkin type $\mathbb{A}_{n}$ and $\mathbb{D}_{n}$. Our proofs are constructive, purely graph theoretical, and almost self-contained in the…
This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
A previously proposed two-step algorithm for calculating the expectation values of Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non- linear equations is…
We discuss applications of the $q$-characters to the computation of the $R$-matrices. In particular, we describe the $R$-matrix acting in the tensor square of the first fundamental representation of E$_8$ and in a number of other cases,…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a…
The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the…