Fractal nil graded Lie superalgebras

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• 作者：Victor Petrogradsky¹ ; ^{petrogradsky@rambler.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16P90 ; 16N40 ; 16S32 ; 17B50 ; 17B65 ; 17B66 ; 17B70
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：466
• 期：Complete
• 页码：229-283
• 全文大小：914 K

文摘

The Grigorchuk and Gupta–Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [31], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [41]. Thus, we have examples of (self-similar) finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [27].

The goal of the present paper is to construct analogues of the Grigorchuk and Gupta–Sidki groups in the world of Lie superalgebras of an arbitrary characteristic. We construct two examples R, Q of finitely generated self-similar Lie superalgebras over a field K   of an arbitrary characteristic and study their properties and properties of their associative hulls. In case $\mathrm{char}K=2$, these examples turn into restricted Lie algebras.

The virtue of the present construction is that the Lie superalgebras have clear monomial bases. These Lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators. The Lie superalgebra R is Z²${\mathbb{Z}}^2$-graded, while Q has a multidegree Z³${\mathbb{Z}}^3$-gradation and a Z²${\mathbb{Z}}^2$-gradation. Both algebras R and Q have similar constructions, computations for R are simpler, but Q enjoys some more specific interesting properties. The Z³${\mathbb{Z}}^3$-components of Q lie inside an elliptic paraboloid in space, they are at most one-dimensional, thus, the Z³${\mathbb{Z}}^3$-grading of Q is fine. In the Z²${\mathbb{Z}}^2$-gradation of Q, all components Q_nm${\mathbf{Q}}_{nm}$, n,m∈Z$n,m\in \mathbb{Z}$, are infinite dimensional except for Q₀₀={0}${\mathbf{Q}}_{00}=\{0\}$. The Z²${\mathbb{Z}}^2$-gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras Q=Q₊⊕Q₋$\mathbf{Q}={\mathbf{Q}}_{+}\oplus {\mathbf{Q}}_{-}$.

In both examples, ad a is nilpotent, a   being even or odd with respect to the Z₂${\mathbb{Z}}_2$-grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta–Sidki groups. In particular, Q is a nil finely-graded Lie superalgebra, which shows that an extension of a theorem due to Martinez and Zelmanov [27] for the Lie superalgebras of characteristic zero is not valid.

Both Lie superalgebras are self-similar, contain infinitely many copies of themselves, let us also call them fractal due to this property.