Geometric Methods in Rep Theory Seminar – Mikhail Kapranov (IPMU)

N-spherical functors and categorification of Euler’s continuants

Abstract. Euler’s continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction in terms of its entries. Remarkably, they make an appearance in the very foundations of category theory: in the formalism of adjoint functors. More precisely, they upgrade to natural complexes of functors built out of a given functor and its iterated adjoints. Requiring exactness of some of these complexes leads to the concept of an N-spherical functor which specializes to that of an ordinary spherical functor for N=4. Such functors describe N-periodic semi-orthogonal decompositions of (enhanced) triangulated categories. Like ordinary spherical functors, they give rise to interesting self-equivalences. Conceptually, they can be seen as categorification of certain irregular differential equations (polynomial Schroedinger) in the complex plane. Joint work with T. Dyckerhoff, V. Schechtman.

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