On existence of PI-exponents of unital algebras

  • Received: 01 April 2020
  • Primary: 16R10; Secondary: 16P90

  • We construct a family of unital non-associative algebras $ \{T_\alpha\vert\; 2<\alpha\in\mathbb R\} $ such that $ \underline{exp}(T_\alpha) = 2 $, whereas $ \alpha\le\overline{exp}(T_\alpha)\le\alpha+1 $. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $ T_\alpha $ does not exist for any $ \alpha> 2 $. This is the first example of a unital algebra whose PI-exponent does not exist.

    Citation: Dušan D. Repovš, Mikhail V. Zaicev. On existence of PI-exponents of unital algebras[J]. Electronic Research Archive, 2020, 28(2): 853-859. doi: 10.3934/era.2020044

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  • We construct a family of unital non-associative algebras $ \{T_\alpha\vert\; 2<\alpha\in\mathbb R\} $ such that $ \underline{exp}(T_\alpha) = 2 $, whereas $ \alpha\le\overline{exp}(T_\alpha)\le\alpha+1 $. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $ T_\alpha $ does not exist for any $ \alpha> 2 $. This is the first example of a unital algebra whose PI-exponent does not exist.



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