Related papers: Trace map, Cayley transform and LS category of Lie…

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group…

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast…

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free…

We classify the transitive, effective, holomorphic actions of connected complex Lie groups on complex surfaces.

Let $G$ be a finite abelian group, let $E$ be a subset of $G$, and form the Cayley (directed) graph of $G$ with connecting set $E$. We explain how, for various matrices associated to this graph, the spectrum can be used to give information…

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…

A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the…

In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where…

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the…

In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms…

We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the…

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$.…

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete…

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even…

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant…

We associate to each covering map of simple Lie groups a sequence of integers, called the multi-degree of the covering; extend Schubert calculus to evaluate the invariant; and apply the results to solve two outstanding topological problems…

Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.