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The goal of my talk (based on a recent joint paper with Dylan Rupel) is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field F_q and any sequence ii of simple objects in C an element X_{V,ii} of the corresponding algebra P_ii of q-polynomials. If C is hereditary, then the assignment V--> X_{V,ii} is an algebra homomorphism from the Hall-Ringel algebra of C to the q-polynomial algebra P_ii, which generalizes the well-known Feigin homomorphisms from the upper half of a quantum group to various q-polynomial algebras.

If C is the representation category of an acyclic quiver Q and ii is the twice repetition-free source-adapted sequence for Q, then we construct an acyclic quantum cluster algebra on P_ii and prove that the the quantum cluster characters X_{V,ii} for exceptional representations Q give all (non-initial) cluster variables in P_ii. This, in particular, settles an important case of a conjecture by A. Zelevinsky and myself on quantum unipotent cells.