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Quadratic automaton algebras and intermediate growth

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 Added by Natalia Iyudu
 Publication date 2017
  fields
and research's language is English




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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 Grobner 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.



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We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Grobner basis among them are identified. We also give a complete classification of cubic algebras A with Hilbert series $H_A=(1+t)^{-1}(1-t)^{-3}$. These two classes of algebras contain all Artin-Schelter regular algebras of global dimension 3. As far as the latter are concerned, our results extend well-known results of Artin and Schelter by providing a classification up to an algebra isomorphism.
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