On the use of graphs for computing a basis, growth and Hilbert series of associative algebras

Certain graphs are considered that can be assigned to an associative algebra and for which there exists a bijective correspondence between paths in the graph and the basis of the algebra. Algorithms for computing the growth and the Hilbert series of the algebra with the use of its graph are indicated. A class of algebras, called automaton algebras, is introduced, for which the rationality of the Hilbert series and the alternativity of the growth are proved. It is shown that commutative algebras, algebras defined by two quadratic relations, algebras defined by the commutativity condition of some generators, and algebras with a finite Gröbner basis are all automaton algebras.
Bibliography: 14 titles