On Jordan doubles of slow growth of Lie superalgebras

TL;DR

This paper explores the properties of Jordan doubles of Lie superalgebras, demonstrating their diverse Gelfand-Kirillov dimensions, constructing examples with specific gradings and growth rates, and discussing self-similarity and wreath products in superalgebras.

Contribution

It introduces new constructions of Jordan superalgebras with arbitrary Gelfand-Kirillov dimensions and specific gradings, expanding understanding of their growth and structural properties.

Findings

Gelfand-Kirillov dimension of Jordan superalgebras can be any number in {0}∪[1,+∞]

Constructed a Jordan superalgebra with linear growth and non-periodic component dimensions

Discussed self-similarity and wreath product concepts in 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. 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 associative and Jordan algebras, one hasn't an analogue of Bergman's gap $(1,2)$ for the Gelfand-Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebra $\mathbf R$ constructed before, we construct a Jordan superalgebra $\mathbf J={\mathcal Jor}({\mathbf R})$ that is nil finely $\mathbb Z^3$-graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk and Gupta-Sidki groups) of Lie algebras in characteristic zero and Jordan algebras in characteristic not 2. Also, $\mathbf J$ is just infinite but not hereditary just infinite. A similar Jordan superalgebra of slow polynomial growth was constructed before. 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 starting with another example of a Lie superalgebra. We discuss the notion of self-similarity for Lie, associative, Poisson, and Jordan superalgebras. We also discuss the notion of a wreath product in case of Jordan superalgebras.