Theory of homotopes in application to mutually unbiased bases, harmonic analysis on graphs and perverse sheaves

TL;DR

This survey explores the theory of homotopes, their algebraic properties, and diverse applications in areas like graph analysis, perverse sheaves, quantum information, and algebraic decompositions.

Contribution

It introduces the concept of well-tempered elements, studies homotopes via Laplace operators, and connects homotope representations to various mathematical and physical structures.

Findings

Homotopes constructed by Laplace operators are quotients of Temperley-Lieb algebras.

Representations of certain homotopes correspond to perverse sheaves on punctured surfaces.

The theory links to orthogonal decompositions, mutually unbiased bases, and quantum protocols.

Abstract

The paper is the survey of the modern results and applications of the theory of homotopes. The notion of a well-tempered element in an associative algebra is introduced and it is proven that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued t-structure. Hochschild and global dimensions of the homotopes are calculated. The homotopes constructed by generalized Lapalce operators in Poincare groupoids of graphs are studied. It is shown that they are quotients of Temperley-Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2 dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of sl(n, C) into the sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, to generalized Hadamard matrices are discussed.