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

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. first author constructed their analogue case restricted Lie algebras characteristic 2 [50], Shestakov Zelmanov extended this construction to an arbitrary positive [68]. Thus, we have with a nil p-mapping. In zero, similar Jordan do not exist by results Martinez [43] [78]. analogues the world superalgebras characteristic, virtue that is clear monomial bases [51], they slow polynomial growth. As periodicity, Z2-homogeneous elements ad-nilpotent. A recent example superalgebra linear growth, finite width 4, just infinite but hereditary [13]. By examples, extension result for zero valid. Now, construct fractal 3-generated Q over field, which gives rise associative hull A, Poisson P, two J K, latter being factor algebra J. charK≠2, has filtration, associated graded structure such grA≅P, also P admits algebraic quantization using deformed A(t). finely Z3-graded multidegree generators, Z3-graded, while K Z4-graded four generators. our construction, these five We describe multihomogeneous coordinates Q, space as bounded “almost cubic paraboloids”. determine hypersurface R4 bounds monomials K. Constructions paper can be applied (super)algebras studied before obtain well. without unit Po, Jo, Ko direct sums locally nilpotent subalgebras there continuum decompositions. Also, Q=Q0¯⊕Q1¯ superalgebra, so, again shows charK=2, Z4-graded, contrast non-existence (roughly speaking, group) distinct from infinite, infinite. call J, because contain infinitely many copies themselves.