Superalgebras and algebras with involution: classifying varieties of quadratic growth

Abstract

Let $F$ be a field of characteristic zero. By a $\phi$-algebra we mean a superalgebra or an algebra with involution over $F.$ In the last years, the sequence of $\phi$-codimensions ${\{c_n^{\phi }(A)\}}_{n\ge 1}$ of a $\phi$-algebra $A$ has been extensively studied. In this paper, we classify varieties generated by unitary $\phi$-algebras having quadratic growth of $\phi$-codimensions. As a consequence we obtain that a unitary $\phi$-algebra with quadratic growth is $T^{\phi }$-equivalent to a finite direct sum of minimal unitary $\phi$-algebras with at most quadratic growth of the $\phi$-codimensions. In addition, we explicit all quadratic functions describing the $\phi$-codimension sequence of a unitary $\phi$-algebra.

Keywords:

• Algebra with involution
• polynomial growth
• polynomial identity
• superalgebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 16R50
• Secondary 20C30
• 16W10

Acknowledgments

The authors would like to thank the referee for his/her useful suggestions.

Additional information

Funding

D. C. L. Bessades and A. C. Vieira were partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and M. L. O. Santos was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).