Finitely presented quadratic algebras of intermediate growth

In this article, we give two examples of finitely presented quadratic algebras (algebras presented by quadratic relations) of intermediate growth

Quadratic automaton algebras and intermediate growth

We present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Gr\"obner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. This answers a question of Ufnarovski. Another result is a simple example (4 generators and 7 relations) of a quadratic algebra of intermediate growth.Comment: To appear in Journal of Cobinatorial Algebr

Growth of Algebras and Codes

This dissertation is devoted to the study of the growth of algebras and formal languages. It consists of three parts. The first part is devoted to the growth of finitely presented quadratic algebras. The study of these algebras was motivated by the question about the growth types of Koszul algebras which are a special subclass of finitely presented quadratic algebras. We show that there exist finitely presented quadratic algebras of intermediate growth and give two concrete examples of such algebras with their presentations. The second part focuses on the study of the growth of metabelian Lie algebras and their universal enveloping algebras. Our motivation was to construct finitely presented algebras of different intermediate growth types. As an outcome of this investigation we prove that for any d 2 N there exists a finitely presented algebra whose growth function is equivalent to e^n^d=(d+1). The last part focuses on infinite codes over finite alphabets, their properties and growth. A special attention is paid to S-codes, weak S-codes and Markov codes which play an important role in coding theory and ergodic theory. We investigate what types of codes may have maximal growth. Also, we prove that S-codes covering Bernoulli schemes are maximal