Séminaire Arithmétique et Géométrie Algébrique
Categorification of Euler continuants and periodic semi-orthogonal decompositions
Nov. 2022
Intervenant : Mikhail Kapranov
Institution : KAVLI IPMU Tokyo
Heure : 14h00 - 15h00
Lieu : Salle 3L15

A functor $F=F_1$ between triangulated categories often has several iterated adjoints
$F_2=F_1^*, F_3=f_2^*, ..., F_n = F_{n-1}^*$. In this case one can form two complexes of functors
by successively applying the counits $F_i F_{i+1}\to Id$ or units $Id\to F_{i+1}F_i$. These complexes can
be seen as categorical lifting of:

-- Fibonacci numbers;

-- Chebyshev polynomials of the second kind;

-- Euler continuants, i.e., universal numerators or denominators of finite continued fractions with
entries being independent variables.

Requiring the totalizations of such complexes to be equivalences of categories or quasi-isomorphic to zero leads
to a generalization of the concept of a spherical functor. Such generalized spherical functors describe periodic
semi-orthogonal decompositions of enhanced triangulated categories. Joint work in progress
with T. Dyckerhoff and V. Schechtman.

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