Theory of homotopes with applications to mutually unbiased bases, harmonic analysis on graphs, and perverse sheaves. (English. Russian original) Zbl 1484.14033
Russ. Math. Surv. 76, No. 2, 195-259 (2021); translation from Usp. Mat. Nauk 76, No. 2, 3-70 (2021).
Summary: 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.
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 16W99 | Associative rings and algebras with additional structure |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 31C20 | Discrete potential theory |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |
Keywords:
homotope; well-tempered element; orthogonal decomposition of Lie algebra; mutually unbiased bases; quantum protocol; Temperley-Lieb algebra; Poincaré groupoid; generalized Hadamard matrix; Laplace operator on graph; discrete harmonic analysis; perverse sheaves; gluing of t-structuresReferences:
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