Fractal just infinite nil Lie superalgebra of finite width

Abstract

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. As their natural analogues, there are constructions of nil Lie p-algebras over a field of characteristic 2 [40] and arbitrary positive characteristic [52]. In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [32].The second author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that the Lie superalgebras have clear monomial bases [41]. That Lie superalgebras have slow polynomial growth and are graded by multidegree in the generators. In particular, a self-similar Lie superalgebra Q is Z3-graded by multidegree in 3 generators, its Z3-components lie inside an elliptic paraboloid in space, the components are at most one-dimensional, thus, the Z3-grading of Q is fine. An analogue of the periodicity is that homogeneous elements of the grading Q=Q0¯⊕Q1¯ are ad-nilpotent. In particular, Q is a nil finely graded Lie superalgebra, which shows that an extension of the mentioned result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid. But computations with Q are rather technical.In this paper, we construct a similar but simpler and “smaller” example. Namely, we construct a 2-generated fractal Lie superalgebra R over arbitrary field. We find a clear monomial basis of R and, unlike many examples studied before, we find also a clear monomial basis of its associative hull A, the latter has a quadratic growth. The algebras R and A are Z2-graded by multidegree in the generators, positions of their Z2-components are bounded by pairs of logarithmic curves on plane. The Z2-components of R are at most one-dimensional, thus, the Z2-grading of R is fine. As an analogue of periodicity, we establish that homogeneous elements of the grading R=R0¯⊕R1¯ are ad-nilpotent. In case of N-graded algebras, a close analogue to being simple is being just infinite. Unlike previous examples of Lie superalgebras, we are able to prove that R is just infinite, but not hereditary just infinite. Our example is close to the smallest possible example, because R has a linear growth with a growth function γR(m)≈3m, as m→∞. Moreover, R is of finite width 4 (charK≠2). In case charK=2, we obtain a Lie algebra of width 2 that is not thin.Thus, we have got a more handy analogue of the Grigorchuk and Gupta-Sidki groups. The constructed Lie superalgebra R is of linear growth, of finite width 4, and just infinite. It also shows that an extension of the result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid.