Reconhecimento polinomial de álgebras cluster de tipo finito - Dias, Elisângela SIlva (2015)
Cluster algebras form a class of commutative algebra, introduced at the beginning of the millennium by Fomin and Zelevinsky. They are defined constructively from a set of generating variables (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Since its inception, the theory of cluster algebras found applications in many areas of science, specially in mathematics. In this thesis, we study, with computational focus, the recognition of cluster algebras of finite type. In 2006, Barot, Geiss and Zelevinsky showed that a cluster algebra is of finite type whether the associated graph is cyclically oriented, i.e., all chordless cycles of the graph are cyclically oriented, and whether the skew-symmetrizable matrix associated has a positive quasi-Cartan companion. At first, we studied the two topics independently. Related to the first part of the criteria, we developed an algorithm that lists all chordless cycles (polynomial on the length of those cycles) and another that checks whether a graph is cyclically oriented and, if so, list all their chordless cycles (polynomial on the number of vertices). Related to the second part of the criteria, we developed some theoretical results and we also developed a polynomial algorithm that checks whether a quasi-Cartan companion matrix is positive. The latter algorithm is used to prove that the problem of deciding whether a skew-symmetrizable matrix has a positive quasi-Cartan companion for general graphs is in NP class. We conjecture that this problem is in NP-complete class.We show that the same problem belongs to the class of polynomial problems for cyclically oriented graphs and, finally, we show that deciding whether a cluster algebra is of finite type also belongs to this class.