CALDERO-CHAPOTON ALGEBRAS.

Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of cluster algebras of Dynkin type, we associate a Caldero-Chapoton algebra A_A to any (possibly infinite-dimensional) basic algebra A. By definition, A_A is (as a vector space) generated by the Caldero-Chapoton functions C_A(M) of the decorated representations M of A. If A = P (Q, W) is the Jacobian algebra defined by a 2-acyclic quiver Q with non-degenerate potential W, then we have A_Q Ç A_A Ç A_Q^up, where A_Q and A_Q^up are the cluster algebra and the upper cluster algebra associated to Q. The set B_A of generic Caldero-Chapoton functions is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra P (Q,W) and was introduced by Geiss, Leclerc and Schroer. 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. We define B_A for arbitrary A, and we conjecture that B_A is a basis of the Caldero-Chapoton algebra A_A. Thanks to the decomposition theorem, all elements of BA can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. [ABSTRACT FROM AUTHOR]