Resumen de Quantum cluster characters of Hall algebras

Arkady Berenstein, Dylan Rupel

• The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field Fq and any sequence i of simple objects in C the element XV,i of the corresponding algebra PC,i of q-polynomials. We prove that if C was hereditary, then the assignments V → XV,i define algebra homomorphisms from the (dual) Hall–Ringel algebra of C to the PC,i, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q, d) and i = (i0, i0), where i0 is a repetition-free source-adapted sequence, then we prove that the i-character XV,i equals the quantum cluster character XV introduced earlier by the second author in Rupel (Int Math Res Not 14:3207–3236, 2011; Quantum cluster characters of valued quivers, arXiv:1109.6694). Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper Berenstein and Zelevinsky (Adv Math 195(2):405–455, 2005) of the first author with A. Zelevinsky for such quantum unipotent cells. As a by-product, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in Berenstein and Zelevinsky (Int Math Res Not. doi:10.1093/imrn/rns268, 2014).