Quantum cluster characters of Hall algebras

TL;DR

This paper introduces a generalized quantum cluster character for Hall algebras, establishing algebra homomorphisms and connecting to quantum cluster structures, thereby proving a conjecture and constructing a quantum twist.

Contribution

It defines a new quantum cluster character for Hall algebras, generalizes Feigin homomorphisms, and proves a conjecture on quantum unipotent cells.

Findings

Quantum cluster characters coincide with earlier definitions in certain cases.

Established a quantum cluster structure on quantum unipotent cells.

Constructed a quantum twist preserving the triangular basis.

Abstract

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 FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that if C was hereditary, then the assignments V-> X_{V,ii} define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, 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 ii=(ii_0,ii_0), where ii_0 is a repetition-free source-adapted sequence, then we prove that the ii-character X_{V,ii} equals the quantum cluster character X_V introduced earlier by the second author in [29] and [30]. 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 [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6].