Related papers: The centre of the extended Haagerup subfactor has …

We consider the category of smooth $W(k)[GL_n(F)]$-modules, where F is a p-adic field and k is an algebraically closed field of characteristic l different from p. We describe a factorization of this category into blocks, and show that the…

Just infinite algebras have been considered from various perspectives; a common thread in these treatments is that the notion of just infinite is an extension of the notion of simple. We reinforce this generalization by considering some…

In this paper, we show the existence of a near-group category of type $\mathbb{Z} / 4\mathbb{Z} \times \mathbb{Z} / 4\mathbb{Z}+16$ and compute the modular data of its Drinfeld center. We prove that a modular data of rank $10$ can be…

Given a metric space, the $(k,z)$-clustering problem consists of finding $k$ centers such that the sum of the of distances raised to the power $z$ of every point to its closest center is minimized. This encapsulates the famous $k$-median…

We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial…

Three types of numerical data are provided for simple Lie groups of any type and rank. This data is indispensable for Fourier-like expansions of multidimensional digital data into finite series of $C-$ or $S-$functions on the fundamental…

We construct a Cartesian product G x H for finite simple graphs. It satisfies the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial…

This paper is mainly concerned with identities like \[ (x_1^d + x_2^d + \cdots + x_r^d) (y_1^d + y_2^d + \cdots y_n^d) = z_1^d + z_2^d + \cdots + z_n^d \] where $d>2,$ $x=(x_1, x_2, \dots, x_r)$ and $y=(y_1, y_2, \dots, y_n)$ are systems of…

For simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the Feigin-Frenkel centre. These formulas make use of the symmetrisation map as well as some well-chosen symmetric invariants of $\mathfrak…

Conformal field theory (CFT) in two dimensions provide a rich source of subfactors. The fact that there are so many subfactors coming from CFT have led people to conjecture that perhaps all finite depth subfactors are related to CFT. In…

We generalize the Pierce representation theorem for (commutative) rings with unit to other algebraic categories with Definable Factor Congruences by using tools from topos theory. Of independent interest, we prove that an algebraic category…

Let $R$ be a finite commutative local principal ring, and let $H(R)$ denote the corresponding quaternion ring. We show that an element of $H(R)$ is a product of idempotents if and only if it can be expressed as a product of two idempotents.…

A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We…

Let $\mathcal{C}$ be a self-dual spherical fusion categories of rank $4$ with non-trivial grading. We complete the classification of Grothendieck ring $K(\mathcal{C})$ of $\mathcal{C}$; that is, we prove that $K(\mathcal{C})\cong…

We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.

We give characterizations of the center, of conjugated and of commuting elements in a fundamental group of a graph of group. We deduce various results : on the one hand we give a sufficient condition for the center, the centralizers, and…

The core of a graph is the smallest graph (in terms of number of vertices) to which it is homomorphically equivalent. The question of the possible order of the core of the tensor product (also known as categorical, Heidetnemi or direct…

For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $\gamma_{\text{cp}}\left( G\right) $ to be the smallest number…

We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup-Izumi family of Q-systems. For example, we construct…

Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over $\Q[\xi]$ in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and…