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

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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 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 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.

Sobre autores

Alexey Bondal

Steklov Mathematical Institute of Russian Academy of Sciences; Moscow Institute of Physics and Technology (National Research University); HSE University; Kavli Institute for the Physics and Mathematics of the Universe

Email: bondal@mi-ras.ru
Doctor of physico-mathematical sciences

Il'ya Zhdanovskiy

Moscow Institute of Physics and Technology (National Research University); Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)

Email: ijdanov@mail.ru
Candidate of physico-mathematical sciences

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