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Differential identities and varieties of almost polynomial growth

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Abstract

Let \({\cal V}\) be an L-variety of associative L-algebras, i.e., algebras where a Lie algebra L acts on them by derivations, and let \(c_n^L\left({\cal V} \right),\,\,n \ge \,1\), be its L-codimension sequence. If \({\cal V}\) is generated by a finite-dimensional L-algebra, then such a sequence is polynomially bounded only if \({\cal V}\) does not contain UT2, the 2 × 2 upper triangular matrix algebra, with trivial L-action, and UT ε2 where L acts on UT2 as the l-dimensional Lie algebra spanned by the inner derivation ε induced by ε11. In this paper we completely classify all the L-subvarieties of varL(UT2) and varL(UT ε2 ) by giving a complete list of finite-dimensional L-algebras generating them.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via Archirafi 34, 90123, Palermo, Italy

    Fabrizio Martino

  2. Departamento de Matemática, CMUC, Universidade de Coimbra, 3004-504, Coimbra, Portugal

    Carla Rizzo

Authors
  1. Fabrizio Martino
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  2. Carla Rizzo
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Corresponding author

Correspondence to Fabrizio Martino.

Additional information

This work was partially supported by the Centre for Mathematics of the University of Coimbra — UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

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Martino, F., Rizzo, C. Differential identities and varieties of almost polynomial growth. Isr. J. Math. 254, 243–274 (2023). https://doi.org/10.1007/s11856-022-2396-1

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  • DOI: https://doi.org/10.1007/s11856-022-2396-1

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