Descrição: Given an associative ring of ${\mathbb Z}_2^n$-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is $b_n= n+\lfloor n/2\rfloor+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 ${\mathbb Z}_2^n\times {\mathbb Z}_2^n\rightarrow {\mathbb Z}_2$ mappings, are defined by consistent sets of commutators\slash 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 particles (bosons, parabosons, fermions, parafermions). As a consequence, the assignment of certain ``marked" operators to a given graded sector is a further mechanism to induce inequivalent graded Lie (super)algebras (the basic examples of quaternions, split-quaternions and biquaternions illustrate these features). \\
As a first application we construct ${\mathbb Z}_2^2$ and ${\mathbb Z}_2^3$-graded quantum Hamiltonians which respectively admit $b_2=4$ and $b_3=5$ inequivalent multiparticle quantizations
(the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). The extension to ${\mathbb Z}_2^n$-graded quantum Hamiltonians for $n>3$ is immediate.\\
As a main physical application we prove that the ${\cal N}$-extended, one-dimensional supersymmetric and superconformal quantum mechanics, for ${\cal N}=1,2,4,8$, are respectively described by $s_{\cal N}=2,6,10,14 $ alternative formulations based on the inequivalent graded Lie (super)algebras. The $s_{\cal N}$ numbers correspond to all possible ``statistical transmutations" of a given set of supercharges
which, for ${\cal N}=1,2,4,8$, are accommodated into a ${\mathbb Z}_2^n$-grading with $n=1,2,3,4$ (the identification is ${\cal N}= 2^{n-1}$).\\
In the simplest ${\cal N}=2$ setting (the $2$-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with $sl(2|1)$ spectrum-generating superalgebra), the ${\mathbb Z}_2^2$-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics. |
