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. 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.