Theory of homotopes with applications to mutually unbiased bases, harmonic analysis on graphs, and perverse sheaves

A. I. Bondal ; I. Yu. Zhdanovskiy

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 195-259

Résumé

This paper is a survey of contemporary results and applications of the theory of homotopes. The notion of a well-tempered element of an associative algebra is introduced, and it is proved that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued t-structure. The Hochschild and global dimensions of homotopes are calculated in the case of well-tempered elements. The homotopes constructed from generalized Laplace operators in Poincaré 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 the Lie algebras $\operatorname{sl}(n,\mathbb{C})$ into a sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, and to generalized Hadamard matrices are discussed. Bibliography: 56 titles.

Détail

Keywords: well-tempered element, mutually unbiased bases, Temperley–Lieb algebra, Poincaré groupoid, generalized Hadamard matrix, Laplace operator on a graph, discrete harmonic analysis, perverse sheaves, gluing of t-structures.
Mots-clés : homotope, orthogonal decomposition of a Lie algebra, quantum protocol