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On Jordan doubles of slow growth of Lie superalgebras

  • Victor PetrogradskyEmail author
  • I. P. Shestakov
Article
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Abstract

To an arbitrary Lie superalgebra L we associate its Jordan double \({\mathcal Jor}(L)\), which is a Jordan superalgebra. This notion was introduced by the second author before (Shestakov in Sib Adv Math 9(2):83–99, 1999). Now we study further applications of this construction. First, we show that the Gelfand–Kirillov dimension of a Jordan superalgebra can be an arbitrary number \(\{0\}\cup [1,+\,\infty ]\). Thus, unlike the associative and Jordan algebras (Krause and Lenagan in Growth of algebras and Gelfand–Kirillov dimension, AMS, Providence, 2000; Martinez and Zelmanov in J Algebra 180(1):211–238, 1996), one hasn’t an analogue of Bergman’s gap (1, 2) for the Gelfand–Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebra \({\mathbf {R}}\) of de Morais Costa and Petrogradsky (J Algebra 504:291–335, 2018), we construct a Jordan superalgebra \({\mathbf {J}}={\mathcal Jor}({{\mathbf {R}}})\) that is nil finely \({\mathbb {Z}}^3\)-graded (moreover, the components are at most one-dimensional), the field being of characteristic not 2. This example is in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk and Gupta–Sidki groups) of Lie algebras in characteristic zero (Martinez and Zelmanov in Adv Math 147(2):328–344, 1999) and Jordan algebras in characteristic not 2 (Zelmanov, E., A private communication). Also, \({\mathbf {J}}\) is just infinite but not hereditary just infinite. A similar Jordan superalgebra of slow polynomial growth was constructed before Petrogradsky and Shestakov (Fractal nil graded Lie, associative, poisson, and Jordan superalgebras. arXiv:1804.08441, 2018). The virtue of the present example is that it is of linear growth, of finite width 4, namely, its \(\mathbb N\)-gradation by degree in the generators has components of dimensions \(\{0,2,3,4\}\), and the sequence of these dimensions is non-periodic. Third, we review constructions of Poisson and Jordan superalgebras of Petrogradsky and Shestakov (2018) starting with another example of a Lie superalgebra introduced in Petrogradsky (J Algebra 466:229–283, 2016). We discuss the notion of self-similarity for Lie, associative, Poisson, and Jordan superalgebras. We also suggest the notion of a wreath product in case of Jordan superalgebras.

Keywords

Growth Self-similar algebras Nil-algebras Graded algebras Lie superalgebra Poisson superalgebras Jordan superalgebras Wreath product 

Mathematics Subject Classification

16P90 16N40 16S32 17B65 17B66 17B70 17A70 17B63 17C10 17C50 17C70 

Notes

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BrasiliaBrasiliaBrazil
  2. 2.Instituto de Mathemática e EstatísticaUniversidade de Saõ PauloSaõ PauloBrazil

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