  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].
