Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

Abstract

We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-graded, while $\mathbf J$, $\mathbf K$ are $\mathbb{Z}^4$-graded. These five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of $\mathbf Q$, $\mathbf A$, $\mathbf P$ in space as bounded by "almost cubic paraboloids". A similar hypersurface in $\mathbb R^4$ bounds monomials of $\mathbf J$, $\mathbf K$. Constructions of the paper can be applied to Lie superalgebras studied before and get Poisson and Jordan superalgebras as well. The algebras ${\mathbf Q}$, ${\mathbf A}$, and the algebras without unit $\mathbf P^o$, $\mathbf J^o$, $\mathbf K^o$ are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, $\mathbf Q=\mathbf Q_{\bar 0}\oplus \mathbf Q_{\bar 1}$ is a nil graded Lie superalgebra. In case $\mathrm{char}\, K=2$, $\mathbf Q$ has a structure of a restricted Lie algebra with a nil $p$-mapping. The Jordan superalgebra $\mathbf K$ is just infinite nil finely $\mathbb Z^4$-graded, while such examples (say, analogues of the Grigorchuk group) of Lie and Jordan algebras in characteristic zero do not exist. We call $\mathbf Q$, $\mathbf A$, $\mathbf P$, $\mathbf J$, $\mathbf K$ fractal because they contain infinitely many copies of themselves.