I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such approach is bound to fail.


Motivation
----------


Let's consider algebra $gl\_n$ with ''bosonic'' generators $B\_{ij}$:
$$[B\_{ij},B\_{kl}]=\delta\_{jk}B\_{il}-\delta\_{li}B\_{jk}$$
They can be realized as bosonic differential operators $B\_{ij}=x\_i\partial\_j$ in commuting variables $x\_i x\_j = x\_j x\_i$


There is a known way to extend this construction to the so-called supersymmetric ($\mathbb{Z}\_2$) case by adding ''fermionic'' generators $F\_{rs}$ with grading $\exp(\frac{2\pi i}{2})$, which can be realized with the help of Grassmanian variables $\psi\_i$, $\psi\_i \psi\_j = - \psi\_j \psi\_i$. These generators satisfy similar relations, but commutation law $[,]$ is turned into anticommutation law $\{,\}$. The pattern is $[B,B]\sim B$, $\{F,F\}\sim B$, $[B,F]\sim F$.


Now I want to generalize it to $\mathbb{Z}\_3$ case.


My approach
-----------


consists in introducing another set of generators $C$ and reassigning the gradings: $|B| = 1$, $|F| = \exp(\frac{2\pi i}{3})$, $|C| = \exp(\frac{4\pi i}{3})$ so that the following pattern will take place: $[B,B]\sim B$, $[B,F]\sim F$, $[B,C]\sim C$, $[F,F]\sim C$, $[F,C]\sim B$, $[C,C]\sim F$. Here instead of usual commutators I mean suitably redefined $\mathbb{Z}\_3$-graded commutators.


The problem
-----------


is that it doesn't work. Indeed, in order to define commutators in this way I need for elements to have commutation relations with cubic roots of $1$. But it isn't possible because if all my variables are similar to each other with the only exception of grading, then, for example, $\psi c = f(\psi,c) c \psi = f(\psi,c) f(c,\psi) \psi c$. Hence, since $f(\psi,c) = f(c,\psi)$ this factor must be a quadratic root of $1$.


The only way out that I could think of was introducing more (I started with two) sets variables which kind of live on a quantum plane: $x\_i^{1} x\_j^{2} = q x\_j^{2} x\_i^{1}$. I could write some algebra, but it looked very messy and is probably wrong.


The question
------------


is whether similar constructions of $\mathbb{Z}\_n$ supersymmetry exist in literature or are easy to exclude on some general grounds.


References where people do something else
-----------------------------------------


There's something called ["color Lie algebras"](https://arxiv.org/abs/math/0407165v5), but I haven't been able to fully digest their ideas.


Richard Kerner et. al. have [considered](https://arxiv.org/abs/hep-th/9607143) a similar problem and solved it by embracing the realm of ternary structures. Is there any hope to stay binary in this brave new world?


Edit concerning explicit expressions for commutators
----------------------------------------------------


I was trying to make use of the following realization of $q(2)$-algebra (belonging to the strange series q(n)) consisting of bosonic $B\_{ij}=x\_i \partial\_{x\_j}+\psi\_i \partial\_{\psi\_j}$ and fermionic $F\_{ij} = x\_i \partial\_{\psi\_j}+\psi\_i \partial\_{x\_j}$, $i,j=1,2$. One can check that these generators indeed form q(2) with signs in commutators naturally appearing according to the degree of variables.


Now, for $\mathbb{Z}\_n$ case I'm going to change notation to make the motivation more visible and set $G\_{ij}^k \equiv x\_i^l \partial\_{x\_j^{l-k}}$ be generators of degree $k$ in $\mathbb{Z}\_n$ commuting variables $x\_i^l$ of degree $l$.


I use summation over repeating indices so that, for example, $\mathbb{Z}\_3$ case looks like $B\_{ij} = x\_i \partial\_{x\_j}+\psi\_i \partial\_{\psi\_j}+c\_i \partial\_{c\_j}$, $F\_{ij} = \psi\_i \partial\_{x\_j}+c\_i \partial\_{\psi\_j}+x\_i \partial\_{c\_j}$, $C\_{ij} = c\_i \partial\_{x\_j}+x\_i \partial\_{\psi\_j}+\psi\_i \partial\_{c\_j}$.


Setting the commutation rules between variables $x\_i^l x\_j^k = \tilde{g}(i,l;j,k) x\_j^k x\_i^l$ I immediately descend to the "non-quantum case" (in the sense described above) and set $\tilde{g}(i,l;j,k)\equiv g(l,k)$.


Then I use the following commutation relations
$$ [G\_{ij}^r, G\_{kl}^p] \equiv G\_{ij}^r G\_{kl}^p - \alpha(r,p) G\_{kl}^p G\_{ij}^r$$
with the goal of determining $\alpha(r,s)$ from the condition of absence of second derivatives:


$$ [G\_{ij}^r, G\_{kl}^p] = x\_i^s \partial\_{x\_j^{s-r}} x\_k^q \partial\_{x\_l^{q-p}} - \alpha(r,p) x\_k^q \partial\_{x\_l^{q-p}} x\_i^s \partial\_{x\_j^{s-r}} = g(r-s,q) g(s,q) g(r-s, p-q) x\_k^q x\_i^s \partial\_{x\_l^{q-p}}\partial\_{x\_j^{s-r}} - \alpha(r,p) g(p-q,s)x\_k^q x\_i^s \partial\_{x\_l^{q-p}}\partial\_{x\_j^{s-r}}$$


so that $\alpha(r,p) = \frac{g(r-s,q) g(s,q) g(r-s, p-q)}{g(p-q,s)}$ for every $s$ and $q$ (which is itself a condition on $g(s,q)$).


However the solution I had in mind while writing this, namely, $g(r,s) = \exp(\frac{2\pi i rs}{n})$ doesn't work here, since $\alpha(r,p) = \exp(\frac{2\pi i}{n}(rp+sq-sp-ps-qs))$ depends on $s$ and $q$. This problem originates in the fact that $x\_i^r x\_j^s =g(r,s) x\_j^s x\_i^r$ with $g(r,s)$ taking values in the $n$-th root of $1$ is incompatible with abelian $g(r,s)$, since $g(r,s)g(s,r)=1$, as follows from $x\_i^r x\_j^s =g(r,s) x\_j^s x\_i^r =g(r,s) g(s,r) x\_i^r x\_j^s$.


It's probably not very comprehensible right now, but I'm not sure how to say it the right way.



Is there Z_n graded supersymmetry?
The realizations (of an algebra through another algebra) you are speaking about are actually homomorphisms. And as such they should map between algebraic structures of the same kind: that is from algebras to algebras, from Lie alg to Lie alg, from graded algebras to graded algebras (graded by the same group), etc


Since you are considering Lie superalgebras and realize them through boson-fermion operators, what you are actually doing is to consider the algebra mixing the bosonic/fermionic degrees of freedom as a superlagebra: this means that you consider it eqquiped with a $\mathbb{Z}\_2$-grading and you pick the unique color function available (unique because there is ony a single bicharacter of the $\mathbb{Z}\_2$ group). This is the function:
$\theta:\mathbb{Z}\_2\times\mathbb{Z}\_2\rightarrow\mathbb{C}^\*$ explicitly given by $\theta(a,b)=(-1)^{\deg a\deg b}$, where $a,b\in\mathbb{Z}\_2$ and $\deg=0,1$ depending on whether the corresponding element is even or odd. This bicharacter actually determines the exact form of the brackets in the LS; that is whether they are commutators ($\theta(a,b)=1$) or anticommutators ($\theta(a,b)=-1$).


If you go to more general gradings (that is $\mathbb{Z}\_3$ which interests you -but it could even be an arbitrary finite, abelian group $\mathbb{G}$) then the "bracket" will generally be defined by $[a,b]=ab-\theta(a,b)ba$, where $\theta:\mathbb{G}\times\mathbb{G}\rightarrow\mathbb{C}^\*$ so essentially the color function (take it as a synonym for a bicharacter in this setting) is what determines the form of the bracket (in higher grading groups with more bicharacters available, the bracket need not be an (anti)commutator, it can have a more general form). $\theta$ also determines the behaviour of the multiplication in the tensor product algebras, so we speak about braided, graded tensor products but this is possibly another story.


If you are interested in these points of view, you coud take a look (excuse me in advance for the self-citation but i think it is relevant here) at:


* [Super-Hopf realizations of Lie Superalgebras: Braided Paraparticle extensions of the Jordan-Schwinger map](https://aip.scitation.org/doi/abs/10.1063/1.3473853)
* [Gradings, Braidings, Representations, Paraparticles: some open problems](https://www.mdpi.com/2075-1680/1/1/74) (see expecially the discussion of p. 80-81 and section 5.1)


The first paper mostly reviews the relevant notions while the second one utilizes realizations in a way which -i think- is close to what you are doing. The difference is that i am using (in this references) "bigger" algebras than the supersymmetric mixture of bosons and fermions you are considering. I am utilizing algebras which mix parabosonic and/or parafermionic generators (your algebras can be recovered as quotients of the later). This gives the opportunity to consider other grading groups (i am mainly focusing to the case of a $\mathbb{Z}\_2\times\mathbb{Z}\_2$ grading group and its colors or bicharacters if you prefer). It seems that such a line of investigation presents interest from both the mathematics and the physics point of view. In:


* [$\mathbb{Z}\_2\times\mathbb{Z}\_2$-graded parastatistics in multiparticle quantum Hamiltonians, F. Topan](http://cbpfindex.cbpf.br/publication_pdfs/notasDeFisica_2021-03-02-20-29-25bm90YXNEZUZpc2ljYQ==.pdf)


the author seems to pursue further similar ideas and methods (focusing more on the physics point of view) while i (together with a couple of colleagues) have some older work on further applications of the $\mathbb{Z}\_2\times\mathbb{Z}\_2$-graded, color Lie algebras (focusing more on the representation-theoretic point of view: that is on the construction of representations of Lie superalgebras through paraparticle realisations). I can send you the relevant references if you are interested in these.


I am not sure if the above shed some light in your questions and your research but i hope you'll find something of interest in these.


**Edit:** Some more references which revolve around $\mathbb{Z}\_n$-graded symmetries (generalizing thus the supersymmetric case mentioned in the OP) and focusing on the physics side, are:


* [$\mathbb{Z}\_n$-Graded Topological Generalizations of Supersymmetry and Orthofermion Algebra, A Mostafazadeh](https://arxiv.org/pdf/hep-th/0303094.pdf)
* [On a $\mathbb{Z}\_2^n$-graded version of supesymmetry, A J Bruce](https://www.researchgate.net/profile/Andrew-Bruce-5/publication/329525569_On_a_mathbbZ_2n-Graded_Version_of_Supersymmetry/links/5c0f67cc4585157ac1b93c11/On-a-mathbbZ-2n-Graded-Version-of-Supersymmetry.pdf)


