Given an associative ring of Z2nβ-graded operators, the number of
inequivalent brackets of Lie-type which are compatible with the grading and
satisfy graded Jacobi identities is bnβ=n+βn/2β+1. This follows
from the Rittenberg-Wyler and Scheunert analysis of "color" Lie (super)algebras
which is revisited here in terms of Boolean logic gates. The inequivalent
brackets, recovered from Z2nβΓZ2nββZ2β mappings, are
defined by consistent sets of commutators/anticommutators describing particles
accommodated into an n-bit parastatistics (ordinary bosons/fermions
correspond to 1 bit). Depending on the given graded Lie (super)algebra, its
graded sectors can fall into different classes of equivalence expressing
different types of (para)bosons and/or (para)fermions. As a first application
we construct Z22β and Z23β-graded quantum Hamiltonians which
respectively admit b2β=4 and b3β=5 inequivalent multiparticle quantizations
(the inequivalent parastatistics are discriminated by measuring the eigenvalues
of certain observables in some given states). As a main physical application we
prove that the N-extended, 1D supersymmetric and superconformal quantum
mechanics, for N=1,2,4,8, are respectively described by sNβ=2,6,10,14
alternative formulations based on the inequivalent graded Lie (super)algebras.
These numbers correspond to all possible "statistical transmutations" of a
given set of supercharges which, for N=1,2,4,8, are accommodated into a
Z2nβ-grading with n=1,2,3,4 (the identification is N=2nβ1). In the
simplest N=2 setting (the 2-particle sector of the de DFF deformed
oscillator with sl(2β£1) spectrum-generating superalgebra), the Z22β-graded
parastatistics imply a degeneration of the energy levels which cannot be
reproduced by ordinary bosons/fermions statistics.Comment: 57 pages, 16 figure