# Gradings, Braidings, Representations, Paraparticles: Some Open Problems

## Abstract

**:**

## 1. Introduction

_{B}and the “free” parafermionic P

_{F}algebras. We are going to call these representations Fock-like due to the fact that they are constructed as generalizations of the usual symmetric Fock spaces of the Canonical Commutation relations (CCR) and the antisymmetric Fock spaces of the Canonical Anticommutation Relations (CAR), leading to generalized versions of the Bose-Einstein and the Fermi-Dirac statistics. In [5] it is shown that the parafermionic Fock-like spaces lead us to a direct generalization of the Pauli exclusion principle. The authors further prove that these representations are parametrized by a positive integer $p$ or, equivalently, that they are classified by the positive integers. However they did not construct analytical expressions for the action of the generators on the specified spaces, due to the intractable computational difficulties inserted by the complexity of the (trilinear) relations satisfied by the generators of the algebra. Apart from some special cases (i.e., single degree of freedom algebras or order of the representations $p=1$) the problem of constructing explicitly the determined representations remained unsolved for more than 50 years. In the same paper [5], the authors introduced a couple of interacting paraparticle algebras mixing parabosonic and parafermionic degrees of freedom: the Relative Parabose Set P

_{BF}, the Relative Parafermi Set P

_{FB}and the straight Commutation and Anticommutation relations, abbreviated SCR and SAR respectively.

_{B}anf the P

_{F}algebras. Employing techniques of induced representations, combined with the well known Lie super-algebraic structure of P

_{B}[9] and Lie algebraic structure of P

_{F}[10,11], together with elements from the representation theory of the (complex) Lie superalgebra $osp(1/2n)$ and the (complex) Lie algebra $so(2n+1)$, they proceed to construct Gelfand-Zetlin bases and calculate the corresponding matrix elements. However, the general cases of P

_{BF}, P

_{FB}, S

_{BF}and S

_{FB}algebras remain still open (even in the case of the finite degrees of freedom).

_{F}had already been known since the time of [10,11]. In the 1980s, the pioneering works of Palev [9] established Lie superalgebraic structures for the Parabosonic algebra P

_{B}and the Relative Parafermi Set P

_{FB}algebra [12,13] as well. The picture expands even more with recent results on the $({Z}_{2}\times {Z}_{2})$-graded $\vartheta $-colored Lie structure of the Relative Parabose Set P

_{BF}algebra [14,15].

#### 1.1. Structure of the Paper

_{B}, the “free” Parafermionic algebra P

_{F}, the Relative Parabose Set algebra P

_{BF}and the Relative Parafermi Set algebra P

_{FB}, the straight Commutation Relations SCR and the straight anticommutation relations SAR. For the sake of completeness, we also review some more or less well known particle algebras of mathematical physics which are directly related to the proposed methods: the Canonical Commutation Relations (CCR), the Canonical Anticommutation Relations (CAR), the symmetric Clifford-Weyl algebra W

_{s}, and the antisymmetric Clifford-Weyl algebra W

_{as}.

## 2. The Algebras, in Terms of Generators and Relations

_{s}corresponds to a “symmetric” or commuting mixture of bosonic and fermionic degrees of freedom. It has been used in [17] for the description of a supersymmetric chain of uncoupled oscillators and it corresponds to the most common choice for combining bosonic and fermionic degrees of freedom. One can find a host of applications, in either problems of physics or mathematics. For instance: in [18,19,20] we have constructions of coherent states in models described by this algebra; in [16,21] it is applied in the Jaynes-Cummings model; and in [22] in a variant of this model. In [23,24,25,26,27,28,29,30] this algebra is used for studying problems of the representation theory of Lie algebras, Lie superalgebras and their deformations. Some authors [23,31] use the terminology symmetric Clifford-Weyl algebra or Weyl superalgebra. The algebra W

_{as}corresponds to an “antisymmetric” or anticommuting mixture of bosons and fermions. Applications—mainly in mathematical problems—can be found in [28,31,32]. Some authors [23,31] refer to this algebra as the antisymmetric Clifford-Weyl algebra.

_{BF}, the Relative Parafermi Set P

_{FB}, the Straight Commutation relations S

_{BF}and the Straight Anticommutation relations S

_{FB}have all been introduced in [5] and constitute different choices of mixing algebraically interacting parabosonic and parafermionic degrees of freedom. Mathematical properties of some of these algebras such as their $G$-graded, $\vartheta $-colored Lie structures and, more generally, their braided group structures have been studied in [12,13] for P

_{FB}and in [14,15,33,34,35] for P

_{BF}. However, the representation theory of these mixed paraparticle algebras remains an almost unexplored subject. To the best of the author’s knowledge, the only works in the bibliography dealing with explicit construction of representations for such algebras has to do with the representations of P

_{BF}

^{(1,1)}i.e., of the Relative Parabose Set algebra combining a single parabosonic and a single parafermionic degree of freedom [36,37,38].

## 3. Braided Group, Ordinary Hopf and $(G,\vartheta )$-Lie Structures for the Mixed Paraparticle Algebras: An Attempt at Classification

#### 3.1. Historical and Conceptual Introduction—Literature Review

**Proposition 3.1:**The following statements are equivalent to each other:

**1.**- $A$ is a $G$-graded algebra (the term superalgebra appears often in physics literature when $G={Z}_{2}$) in the sense that $A={\oplus}_{g\in G}{A}_{g}$ and ${A}_{g}{A}_{h}\subseteq {A}_{gh}$ for any $g,h\in G$.
**2.**- $A$ is a (left) $\u2102G$-module algebra.
**3.**- $A$ is a (right) $\u2102G$-comodule algebra.
**4.**- $A$ is an algebra in the Category ${}_{\u2102G}\mathfrak{M}$ of representations (modules) of the group Hopf algebra $\u2102G$.
**5.**- $A$ is an algebra in the Category ${\mathfrak{M}}^{\u2102G}$ of corepresentations (comodules) of the group Hopf algebra $\u2102G$.

^{*}between $\u2102G$ and its dual Hopf algebra ($\u2102G$)

^{*}(where ($\u2102G$)

^{*}$=Hom$$(\u2102G,\u2102)\cong Map(G,\u2102)={\u2102}^{G}$ as complex vector spaces and with ${\u2102}^{G}$ we denote the complex vector space of the set-theoretic maps from the finite abelian group $G$ to $\u2102$). The essence of the description provided by Proposition 3.1 is that the $G$-grading on the algebra $A$ can be equivalently described as a specific (co)action of the group $G$ (and thus of the group Hopf algebra $\u2102G$) on $A$ i.e., a (co)action which “preserves” the algebra structure of $A$. Such ideas, which provide an equivalent description of the grading of an algebra $A$ by a group $G$ as a suitable (co)action of the group Hopf algebra $\u2102G$ on $A$, are actually not new and already appear in works such as [55,56].

**Proposition 3.2:**The following statements are equivalent to each other:

**1.**- $C$ is a $G$-graded coalgebra (the term supercoalgebra seems also appropriate when $G={Z}_{2}$) in the sense that $\mathsf{\Delta}({C}_{\kappa})\subseteq {\oplus}_{g\in G}{C}_{g}\otimes {C}_{{g}^{-1}\kappa}\equiv {\oplus}_{gh=\kappa}{C}_{g}\otimes {C}_{h}$ for any $g,h,\kappa \in G$ and $\epsilon ({C}_{\kappa})=\left\{0\right\}$ for all $\kappa \ne 1\in G$. ($\mathsf{\Delta}:C\to C\otimes C$ and $\epsilon :C\to \u2102$ are assumed to be the comultiplication and the counity respectively).
**2.**- $C$ is a (left) $\u2102G$-module coalgebra.
**3.**- $C$ is a (right) $\u2102G$-comodule coalgebra.
**4.**- $C$ is a coalgebra in the Category ${}_{\u2102G}\mathfrak{M}$ of representations (modules) of the group Hopf algebra $\u2102G$.
**5.**- $C$ is a coalgebra in the Category ${\mathfrak{M}}^{\u2102G}$ of corepresentations (comodules) of the group Hopf algebra $\u2102G$.

**-**Hopf algebras or $G$

**-**graded, $\vartheta $

**-**braided Hopf algebras (see the relative discussion in [35,47]). In this last case, $\vartheta :G\times G\to {\u2102}^{*}$ stands for a skew-symmetric bicharacter [47] on $G$ (or: commutation factor [61,62,63] or color function [65,66]), which has been shown [47,64] to be equivalent to a triangular universal $R$ -matrix on the group Hopf algebra $\u2102G$. This finally entails [47,48,49,64] a symmetric braiding in the Monoidal Category ${}_{\u2102G}\mathfrak{M}$ of the modules over the group Hopf algebra $\u2102G$.

**-**Hopf algebras belong to the—conceptually wider—class of Braided Groups (in the sense of the braiding described above). Here we use the term “braided group” loosely, in the sense of [48,49]. It is also customary to speak of such structures as Hopf algebras in the braided Monoidal Categories ${}_{\u2102G}\mathfrak{M}$ of representations of $\u2102G$. The following proposition (see [47,48,49]) summarizes various different conceptual understandings of the term $G$-graded, $\vartheta $-braided Hopf algebra (see also the corresponding definitions of [47,48,49,68]).

**Proposition 3.2:**The following statements are equivalent to each other:

**1.**- $H$ is a $G$-graded, $\vartheta $-braided Hopf algebra or a $(G,\vartheta )$-Hopf algebra.
**2.**- $H$ is a Hopf algebra in the braided Monoidal Category ${}_{\u2102G}\mathfrak{M}$ of representations of $\u2102G$.
**3.**- $H$ is a braided group for which the braiding is given by the function $\vartheta :G\times G\to {\u2102}^{*}$.
**4.**- $H$ is simultaneously an algebra, a coalgebra and a $\u2102G$-module, all its structure functions (multiplication, comultiplication, unity, counity and antipode) are $\u2102G$-module morphisms. The comultiplication $\underset{\_}{\mathsf{\Delta}}:H\to H\underset{\_}{\otimes}H$ and the counity $\underset{\_}{\epsilon}:H\to \u2102$ are algebra morphisms in the braided monoidal Category ${}_{\u2102G}\mathfrak{M}$. ($H\underset{\_}{\otimes}H$ stands for the braided tensor product algebra). At the same time, the antipode $S:H\to H$ is a “twisted” or “braided” anti-homomorphism in the sense that $S(xy)=\vartheta (\mathrm{deg}(x),\mathrm{deg}(y))S(y)S(x)$for any homogeneous $x,y\in H$.
**5.**- The $\u2102G$-module $H$ is an algebra in ${}_{\u2102G}\mathfrak{M}$ (equiv.: a $\u2102G$-module algebra) and a coalgebra in ${}_{\u2102G}\mathfrak{M}$ (equiv.: a $\u2102G$-module coalgebra), the comultiplication $\underset{\_}{\mathsf{\Delta}}:H\to H\underset{\_}{\otimes}H$ and the counity $\underset{\_}{\epsilon}:H\to \u2102$ are algebra morphisms in the braided monoidal Category ${}_{\u2102G}\mathfrak{M}$ and at the same time, the antipode $S:H\to H$ is an algebra anti-homomorphism in the braided monoidal Category ${}_{\u2102G}\mathfrak{M}$.

_{F}and parabosonic P

_{B}algebras have been shown to be (see the discussion in the introduction, in Section 2 and also [69,70] for a review) isomorphic to the Universal Enveloping Algebra (UEA) of a Lie algebra and a Lie superalgebra (or: ${\mathbb{Z}}_{2}$-graded Lie algebra) respectively, while the Relative Parabose set algebra P

_{BF}has been shown [14,15] to be isomorphic to the UEA of a (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-graded Lie algebra. At the same time the Relative Parafermi set algebra P

_{FB}has been shown [12,13] to be isomorphic to the UEA of a Lie superalgebra. In [69,71] we have studied the case of P

_{B}, and we establish its braided group structure (here: ${\mathbb{Z}}_{2}$-graded Hopf structure) independently of its ${\mathbb{Z}}_{2}$-graded Lie structure.

#### 3.2. Description of the Problem–Research Objectives

_{BF}: In [34,35] we review P

_{BF}as the UEA of a (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-graded, $\vartheta $-colored Lie algebra (for a specific choice of the commutation factor $\vartheta $ proposed in [14,15]). However, in [37,38] we adopt a different point of view, in which we consider P

_{BF}as a (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-graded associative algebra, with a different (inequivalent) form of the grading i.e., with a different $\u2102$(${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-action. In this last case, the (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-grading is not necessarily associated to some particular color-graded Lie structure. We intend to rigorously investigate further, the following points:

- Given the (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-grading described in [14,15,34,35] we intend to check whether it is compatible with other commutation factors $\vartheta $ (i.e.,: other braidings for the ${}_{\u2102\left({\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\right)}\mathfrak{M}$ Category of modules) than the one presented in these works. In other words, we are going to determine possible alternative braided group structures, corresponding to the single (${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$)-graded structure for P
_{BF}described in the above works. It will also be interesting to examine, which of these alternatives—if any—are directly associated to some particular color-graded Lie structure (directly in the sense that they may stem from the UEA). - We are going to determine possible alternative $G$-gradings for the P
_{BF}, P_{FB}(co)algebras where the group $G$ may either be ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ itself (with some grading inequivalent to the previous, in the sense formerly described) or some other suitable group, for ex. ${\mathbb{Z}}_{2}$ or ${\mathbb{Z}}_{4}$. In each case, we will further investigate the possible braidings (in the sense analyzed in the former paragraph). - We are going to collect the results of the previous two steps and develop Theorems and Propositions which establish the possible braided group structures of P
_{BF}and P_{FB}independently of the possible color-graded Lie structures. For each of the above cases, we intend to explicitly compute: (a) The group action (i.e., the grading); (b) The braiding (i.e., the family of isomorphisms), the commutation factor (i.e., the bicharacter or equiv: the color function), (c) The (quasi)triangular structure (i.e., the $R$-matrix) of the corresponding group Hopf algebra.

**1st Research Objective:**The first problem we intend to investigate is the classification of the gradings induced on the paraparticle (co)algebras (especially on P

_{BF}and P

_{FB}algebras) by small order finite Abelian Groups such as ${Z}_{2},{Z}_{3},{Z}_{4},{Z}_{2}\times {Z}_{2}$ etc. In other words, we intend to classify those group (co)actions which preserve the corresponding (co)algebra structures, turning thus the (co)algebras in $\u2102G$-(co)module (co)algebras.

**2nd Research Objective:**Further, for each of the above gradings we intend to classify the corresponding braided group structures. In other words, we will write down the possible bicharacters of the above groups or equivalently the possible $R$-matrices of the corresponding group Hopf algebras or equivalently the braidings of the corresponding Category ${}_{\u2102G}\mathfrak{M}$ (or ${\mathfrak{M}}^{\u2102G}$) of modules (or comodules). For each one of these braidings, we aim to examine whether or not there are available compatible graded algebraic and coalgebraic structures suitable for producing a braided group.

**3rd Research Objective:**Apply or develop suitable bosonization or bosonization-like techniques to obtain ordinary Hopf structures, with no grading and with trivial braiding, possessing equivalent representation theories.

## 4. An Attempt to Approach the Fock-like Representations for the ${P}_{B},{P}_{F},{P}_{BF},{P}_{FB}$ Algebras Utilizing Their Braided Group Structures

#### 4.1. Conceptual Introduction–Methodological Review

_{B}which has been extensively studied in [69,70,71], and based on it, we develop a ‘‘braided interpretation’’ of the Green ansatz for parabosons. We further develop a method, for employing this braided interpretation in order to construct analytic expressions for the matrix elements of the Fock-like representations of P

_{B}. Concisely, the method consists of the following steps:

- regarding $CAR$ (the usual Weyl algebra or: boson algebra) as a superalgebra with odd generators, and proving that it is isomorphic (as an assoc. superalgebra) to a quotient superalgebra of P
_{B}, - constructing the graded tensor product representations, of (graded) tensor powers of the form $CAR\otimes CAR\otimes \mathrm{...}\otimes CAR$ ($p$-copies),
- pulling back the module structure to a representation of P
_{B}through suitable (homogeneous) homomorphisms of the form ${P}_{B}\to CAR\otimes CAR\otimes \mathrm{...}\otimes CAR$, which are constructed via the braided comultiplication $\mathsf{\Delta}:{P}_{B}\to {P}_{B}\underset{\_}{\otimes}{P}_{B}$ of P_{B}(see [103]), - prove that the ${P}_{B}$-modules thus obtained, are isomorphic (as ${P}_{B}$-modules) to ${\mathbb{Z}}_{2}$-graded tensor product modules, between $p$-copies, of the first ($p=1$) Fock-like representation of ${P}_{B}$,
- prove that the parabosonic $p$-Fock-like module, corresponding to arbitrary value of the positive integer $p$, is contained as an irreducible direct summand of the above constructed ${\mathbb{Z}}_{2}$-graded tensor product representation,
- compute explicitly the action of the P
_{B}generators and the corresponding matrix elements, on the above mentioned $p$-Fock-like modules and finally, - decompose the obtained ${\mathbb{Z}}_{2}$-graded tensor product representations into irreducible components and investigate whether more irreducible summands arise, non-isomorphic to the $p$-Fock-like submodule.

#### 4.2. Description of the Problem–Research Objectives

- We first intend to proceed to the explicit construction of the Fock-like representations in the case of the (inf. deg. of freedom) parabosonic P
_{B}and parafermionic P_{F}algebra following the methodology developed in [103] and outlined above. Starting from the parabosonic algebra, this involves computations of expressions of the following form_{F}, since the corresponding variables ${f}_{{i}_{r}}^{(k)+}=I\otimes I\otimes \mathrm{...}\otimes {f}_{{i}_{r}}^{+}\otimes I\otimes \mathrm{...}\otimes I$ (${f}_{{i}_{r}}^{+}$ is the CAR generator) appear to be commuting (the exact choice of the braiding and the grading depends of course on the results of the previous part of the project). What we are actually describing here, are the steps for the explicit calculation of the action of the generators on the tensor product representations of—suitably—graded versions of CCR and CAR and the subsequent decomposition of these representations in irreducible components. In [103] we have proved that the $p$-Fock-like modules are contained as irreducible factors of such graded, tensor product representations. However, it remains to see whether such decompositions can produce as direct summands or more generally as submodules other non-equivalent representations as well.

_{B}generators respectively and m denotes the number of the generators i.e. the possible values of $i$ and $j$):

- Next, we intend to compare our obtained (according to the above described method) results with those obtained in [6,7,8] (where a totally different approach, based on induced representations and chains of inclusions of Lie superalgebras contained as subalgebras, has been adopted). It is expected that the identification of the representations may lead us to valuable insight, relative to the interrelations between the various, diversified analytical tools used.
- The next step will consist of generalizing the above calculations for the case of the mixed paraparticle algebras P
_{BF}and P_{FB}. The philosophy of the method is based on the same idea: The Fock-like representations of P_{BF}and P_{FB}will be extracted as irreducible submodules arising in the decomposition of the graded tensor product representations of ${W}_{s}$ and ${W}_{as}$. In this case, ${W}_{s}$ is a mixture of commuting (symmetric mixture) bosons and fermions and ${W}_{as}$ a mixture of anticommuting (antisymmetric mixture) of bosonic and fermionic generators (see also [54] § 6.2 pp. 199–207, [31] for more details on the structure of these algebras). Just as the CCR may be considered a graded quotient algebra of P_{B}(see [103]) , and the CAR a graded quotient algebra of P_{F}, in the same spirit we will consider ${W}_{s}$ as a suitable graded quotient of P_{BF}and ${W}_{as}$ as a graded quotient of P_{FB}. These are exactly the algebras we intend to employ, in order to generalize the formerly described method for the case of the mixed paraparticle algebras P_{BF}(Relative Parabose Set algebra) and P_{FB}(Relative Parafermi Set algebra). The results of the previous part of the project (i.e., Section 3.) are expected to lead us in suitable choices for the grading and the braiding of ${W}_{s}$ and ${W}_{as}$ (in the same manner that the results of [69,70,71] led us to the use of odd-bosons in [103]). Finally it is worth mentioning, that the computational problem we expect to reveal here is the development of a suitable multinomial theorem mixing commuting and anticommuting variables.

## 5. A Proposal for the Development of an Algebraic Model for the Description of the Interaction between Monochromatic Radiation and a Multiple Level System

#### 5.1. Review of Recent Work

_{BF}and the Relative Parafermi algebra P

_{FB}such as their gradings, braided group structures, θ-colored Lie structures, their subalgebras, etc. These algebras, constitute paraparticle systems defined in terms of parabosonic and parafermionic generators (or: interacting parabosonic and parafermionic degrees of freedom, in a language more suitable for physicists) and trilinear relations. We have then proceeded in building realizations of an arbitrary Lie superalgebra $L={L}_{0}\oplus {L}_{1}$ (of either fin or infin dimension) in terms of these mixed paraparticle algebras. Utilizing a given ${\mathbb{Z}}_{2}$-graded, finite dimensional, matrix representation of L, we have actually constructed maps of the form $J:L\to gl(m/n)\subset \begin{array}{c}{P}_{BF}\\ {P}_{FB}\end{array}$ from the LS L onto a copy of the general linear superalgebra $gl(m/n)$ isomorphically embedded into either P

_{BF}or into P

_{FB}. These maps have been shown to be graded Hopf algebra homomorphisms or more generally braided group isomorphisms and constitute generalizations and extensions of older results [107]. From the viewpoint of mathematical physics, these maps generalize—in various aspects (see the discussion in [35])—the standard bosonic-fermionic Jordan-Scwinger [108,109] realizations of Quantum mechanics. In [37,38] we have further proceeded in building and studying a class of irreducible representations for the simplest case of the P

_{BF}

^{(1,1)}algebra in a single parabosonic and a single parafermionic degree of freedom (a 4-generator algebra). We have used the terminology “Fock-like representations” because these representations apparently generalize the well known boson-fermion Fock spaces of Quantum Field theory.

_{BF}

^{(1,1)}constitute a family parameterized by the values of a positive integer p. They have the general form ${\oplus}_{n=0}^{p}{\oplus}_{m=0}^{\infty}{V}_{m,n}$ where p is an arbitrary (but fixed) positive integer. The subspaces V

_{m}

_{,n}are 2-dim except for the cases m = 0, n = 0, p, i.e., except the subspaces V

_{0,n}, V

_{m}

_{,0}, V

_{m}

_{,p}which are 1-dim for all values of m and n. These subspaces can be visualized as follows:

#### 5.2. Description of the Problem–Research Objectives

_{b}stands for the energy of any paraboson field quanta (this generalizes the photon, represented by the Weyl algebra part of the usual JC-model), ω

_{f}for the energy gap between the subspaces V

_{m}

_{,n}and V

_{m}

_{,n+1}(this generalizes the two-level atom, represented by the su(2) generators of the usual JC-model) and λ or λ

_{i}(i = 1,2) suitably chosen coupling constants. Notice that ω

_{b}and ω

_{f}might be some functions of m or n or both. The H

_{b}+ H

_{f}part of the above Hamiltonians represents the “field” and the “atom” respectively, while the ${H}_{\mathrm{int}eract}=\frac{\lambda}{2}\left(\left\{{b}^{-},{f}^{+}\right\}+\left\{{b}^{+},{f}^{-}\right\}\right)$, ${H}_{\mathrm{int}eract}^{\ast}={\lambda}_{1}{b}^{-}{f}^{+}+{\lambda}_{2}{f}^{+}{b}^{-}+{\lambda}_{2}^{*}{b}^{+}{f}^{-}+{\lambda}_{1}^{*}{f}^{-}{b}^{+}$ operators “simulate” the “field-atom” interactions causing transitions from any V

_{m}

_{,n}subspace to the subspace V

_{m}

_{–1,n+1}$\oplus $V

_{m}

_{+1,n–1}(absorptions and emissions of radiation). The Fock-like representations, the formulas for the action of the generators and the corresponding carrier spaces, will provide a full arsenal for performing actual computations in the above conjectured Hamiltonian and for deriving expected and mean values for desired physical quantities. A preliminary version of these ideas, for the simplest case of P

_{BF}

^{(1,1)}has already appeared (see the discussion at Section 5 of [37]). The spectrum generating algebra of H may be considered to be either P

_{BF}

^{(1,1)}or P

_{FB}

^{(1,1)}or more generally any other mixed paraparticle algebra whose representations can be directly deduced from those of P

_{BF}

^{(1,1)}or P

_{FB}

^{(1,1)}: Such algebras may be the “straight” Paraparticle algebras $SC{R}^{(1,1)}\cong {{P}_{B}}^{(1)}{\otimes}^{Gr}{{P}_{F}}^{(1)}$ or $SA{R}^{(1,1)}\cong {{P}_{B}}^{(1)}{\otimes}_{gr}{{P}_{F}}^{(1)}$ where ${\otimes}^{Gr}$ and ${\otimes}_{gr}$ stand for braided tensor products for suitable choices of the grading group G and the braiding function θ. More details on the choices of the grading groups and the braiding functions and on the above mentioned isomorphisms will be given in the forthcoming work [110].

_{BF}or P

_{FB}in contrast to the SCR and SAR algebras where only commutation (anticommutation) relations are involved between generators of different “species” indicate that we may expect a more promising simulation of the dynamics by the P

_{BF}or P

_{FB}algebras in conjunction with the “free” Hamiltonian ${H}_{free}$.

## 6. Conclusions

_{s}and W

_{as}for the explicit construction of families of Fock-like representations of the paraparticle algebras. Special attention is paid in the description of unsolved mathematical problems related to the method and dealing with the development of multinomial expansions mixing commuting and anticommuting variables.

## Acknowledgments

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## Appendix: Sketch of the Proof of Proposition 3.2

**Definition A.1**: First of all $C$ being a (right) $H$-comodule coalgebra means that:

**a.**- $C$ is a right $H$-comodule (with the coaction denoted by ${\rho}_{C}$).
**b.**- Its structure maps i.e., the comultiplication ${\Delta}_{C}:C\to C\otimes C$ and the counity ${\epsilon}_{C}:C\to \u2102$, are H-comodule morphisms.

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Kanakoglou, K.
Gradings, Braidings, Representations, Paraparticles: Some Open Problems. *Axioms* **2012**, *1*, 74-98.
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**AMA Style**

Kanakoglou K.
Gradings, Braidings, Representations, Paraparticles: Some Open Problems. *Axioms*. 2012; 1(1):74-98.
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2012. "Gradings, Braidings, Representations, Paraparticles: Some Open Problems" *Axioms* 1, no. 1: 74-98.
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