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Oberseminar WiSe 2011/12 Zusammenfassung des Vortrags von Mikhail Kapranov (Yale, USA)

Arithmetic Hall algebras

The construction of the Hall algebra of an abelian category A is known to produce interesting Hopf algebras of quantum groujp-theoretic nature. One is unually imposes the condition that A has cohomological dimension 1, and two classes of such A have been studied: representations of a quiver and coherent sheaves on a curve X over Fq.

The talk will discuss the third case when X is replaced by the spectrum of the ring of integers in a number field, compactified at the infinity. In this case one has the analog of the Hall algebra consisting of nonholomorphic automorphic forms, and one can describe its structure as a Feigin-Odesskii shuffle algebra. Joint work with O. Schiffmann and E. Vasserot.