N-spherical functors, categorification of Euler's continuants and periodic semi-orthogonal decompositions

The concept of a spherical functor involves the cones of the unit and counit of the adjunction for a pair of adjoint functors. The 2-term complexes formed by the unit and counit are simplest instances of more general complexes of functors which can be seen as categorical liftings of Euler continuants, the universal polynomials expressing the numerator and denominator of a continuous fraction in terms of its coefficients.  Imposing natural conditions on the totalization of these complexes, we get a concept of an N-spherical functor (with the usual case corresponds to N=4). It turns out that such functors lead to N-periodic semi-orthogonal decompositions and provide a generalization of spherical twists. Joint work with T. Dyckerhoff, V. Schechtman.

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