Motivated by the representation theory of quivers with potentials introduced
by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave
explicit formulae for the cluster variables of Dynkin quivers, we associate a
Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra.
By definition, the Caldero-Chapoton algebra is (as a vector space) generated by
the Caldero-Chapoton functions of the decorated representations of the basic
algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a
quiver with potential is closely related to the cluster algebra and the upper
cluster algebra of the quiver. The set of generic Caldero-Chapoton functions,
which conjecturally forms a basis of the Caldero-Chapoton algebra) is
parametrized by the strongly reduced components of the varieties of
representations of the Jacobian algebra and was introduced by Geiss, Leclerc
and Schr\"oer. Plamondon parametrized the strongly reduced components for
finite-dimensional basic algebras. We generalize this to arbitrary basic
algebras. Furthermore, we prove a decomposition theorem for strongly reduced
components. Thanks to the decomposition theorem, all generic Caldero-Chapoton
functions can be seen as generalized cluster monomials. As another application,
we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton
algebras lead to several general conjectures on cluster algebras.Comment: 35 pages. v2: Corrected the definition of a generic E-invariant and
also a short list of minor inaccuracies and typos. Final version to appear in
Transactions AM