On Jordan doubles of slow growth of Lie superalgebras

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.