Related papers: Three lectures on automorphic loops

We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involution, focusing on properties that remain valid in the…

These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local times and Gaussian processes. This is an…

We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this and \'etale cohomology considerations, we list all different isomorphism classes of the corresponding twisted…

We investigate Moufang loops which can be written as the semidirect product of a loop and a group. We also examine a particular class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms…

For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the M\"obius transformations that commute with $f$, and these form a finite subgroup of…

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the…

We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general this is not true for finite left automorphic Moufang loops and for finite automorphic loops.

These are notes based on a course that I gave at the University of Chicago in Fall 2016 on "Loop measures and the loop-erased random walk." This is not intended to be a comprehensive view but rather a personal selection of some key ideas…

This is the first in a series of papers presenting a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. We study the simplest theory of colored scalar…

Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite…

A map on a group into itself is called a local automorphism if at any two points of the group, it can be interpolated by an automorphism of that group. In this paper we investigate the question of how local automorphisms of some classical…

For any code loop $L$, we prove that the half-automorphism group of $L$ is the product of the automorphism group of $L$ by an elementary abelian $2-$group consisting of all half-automorphisms that acts as the identity on a fixed basis.…

This treatise concerns shapes in the sense of constellations of points with various automorphisms quotiented out: continuous translations, rotations and dilations, and also discrete mirror image identification and labelling…

By a map we mean a $2$-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. Automorphism of a map can be thought of as a permutation of the vertices which…

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable $1$-dimensional…

A \emph{loop} $(B,\cdot)$ is a set $B$ together with a binary operation $\cdot$ such that (i) for each $a\in B$, the left and right translation mappings $L_{a}:B\to B: x \mapsto a\cdot x$ and $R_{a}:B\to B: x \mapsto x\cdot a$ are…

Let $F$ be a perfect field and $M^*(F)$ the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then $Aut(M^*(F))$ is equal to $G_2(F) \rtimes Aut(F)$. In particular,…

In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with…

This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and…