On Jordan doubles of slow growth of Lie superalgebras

Author(s)

Petrogradsky, Victor ^[1] ; Shestakov, I. P. ^[2]

Affiliation

^[1] Univ Brasilia, Dept Math, BR-70910900 Brasilia, DF - Brazil

^[2] Univ Sao Paulo, Inst Math & Estat, Caixa Postal 66281, BR-05315970 Sao Paulo - Brazil

• Document type: Journal article
• Source: SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. 13, n. 1, p. 158-176, JUN 2019.
• Web of Science Citations: 0

Abstract

To an arbitrary Lie superalgebra L we associate its Jordan double J or(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]boolean OR {[} 1,+ infinity]. 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 superalgebraR of de Morais Costa and Petrogradsky (J Algebra 504: 291-335, 2018), we construct a Jordan superalgebra J = J or(R) that is nil finely 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, 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 N-gradationbydegree 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. (AU)

• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 16/18068-9 - Combinatorial aspects of Lie Algebras and of noncomutative algebras
• Grantee: Ivan Chestakov
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil