Medical, Pharma, Engineering, Science, Technology and Business

^{1}Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid,
3 Plaza de Ciencias, 28040 Madrid, Spain
**
E-mail:** [email protected]

^{2}Laboratoire de Physique Theorique, CNRS UMR 7085, Universite Louis Pasteur,
3 rue de l'Universite, 67084 Strasbourg Cedex, France
**
E-mail:** [email protected]

**Received Date**: November 19, 2008; **Revised Date: **March 10, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Cliord-like algebras. It is, moreover, shown that color algebras admit realizations as q = 0 quon algebras.

The problem of nding mass formulae for particles belonging to a representation of an interaction group motivated, in the beginning sixties, the eorts to combine interactions with relativistic invariance in a nontrivial way. The well-known obstructions to such a construction [11,12,13] finally led to the supersymmetric schemes. In this sense, two different models unifying internal and space-time symmetries, the conformal superalgebra and the orthosymplectic algebra were proposed [18,19]. The introduction of a grading was soon recognized to be an indispensable requirement to introduce transformations relating states obeying different quantum statistics types. However, a further generalization seemed necessary to clearly distinguish the color and avor degrees of freedom, which were treated in the same way in the two previous models. A first approach to this question was made in [26], where a color algebra based on the nonassociative octonions was proposed. This scheme constituted a mathematical model taking into account the unobservability of quarks and their associated massless color gauge bosons. An alternative construction, preserving the associative framework, was given in [42,43]. These structures motivated by themselves the study of several extensions of Lie algebras, keeping in mind the main properties that made them interesting to describe symmetries of physical phenomena. Among others, the generalizations that have been proven to be physically relevant are color (or graded) Lie algebras [2,3,4,5,6,7,8,9,10,25,28,29,30,31,32,33,42,43,49] and, more recently, Lie algebras of order [22,23,24,34,35,36]. These two types of algebras share some properties, and are based upon a grading by an Abelian group.

Tensor products constitute a natural tool to construct higher-dimensional algebraic objects starting from two given ones, as well as to study their representation theory and the underlying Clebsch-Gordan problem. However, the tensor product of two algebras usually give rise to nontrivial identities that must be satisfied, and often fail to preserve certain key properties (as happens, e.g., for Lie algebras). In order to prevent this situation, generalized tensor products have been developed for various structures, such as groups or Lie (super)algebras. In a more general frame, there is no reason to believe that tensor products of algebras with different structures cannot lead to further interesting structures, possibly preserving some of the main properties of their components. Some attention has been devoted, in this direction, to tensor products of the type , where C is a Clifford algebra C and D a -graded ring of differential operators on a manifold, where the differential operators are interpreted, in some sense, as "quantum mechanical," the classical approximation of which is given by a Poisson bracket. These products suggest a deep relation with the commutation-anticommutation formalism in eld theory [14].

In this paper, we show that color algebras and Lie algebras of order *F* can be unied leading
to some new algebras that we call color Lie (super)algebra of order *F*. Furthermore, we show that
many examples of these algebras can be seen as tensor products of given algebras. In Section 2, we
recall some results concerning the general theory of color algebras, focusing on the isomorphism
between color and graded algebras [1]. We also review the main features of a distinguished
class of F-ary algebra. Section 4 is devoted to giving a unication of both mentioned types of
structures, as well as an adapted decoloration theorem. It turns out that color algebras arise as
tensor products of ordinary noncolor algebras with algebras of Clifford type. In the last section,
we show that all considered types of algebras are strongly related to quon algebras for q = 0. In
particular, "differential" realizations in terms of quon algebras are obtained.

Color Lie (super)algebras, originally introduced in [42,43], can be seen as a direct generalization of Lie (super)algebras. Indeed, the latter are defined through antisymmetric (commutator) or symmetric (anticommutator) products, although for the former the product is neither symmetric nor antisymmetric and is defined by means of a commutation factor. This commutation factor is equal to ±1 for (super)Lie algebras and more general for arbitrary color Lie (super)algebras.

As happened for Lie superalgebras, the basic tool to dene color Lie (super)algebras is a grading determined by an Abelian group. The latter, besides dening the underlying grading in the structure, moreover, provides a new object known as commutation factor.

**Definition 2.1.** Letbe an Abelian group. A commutation factor *N* is a map satisfying the following constraints:

The denition above implies, in particular, the following relations:

(2.1)

where 0 denotes the identity element of . In particular, xing one element of , the induced mapping denes a homomorphism of groups.

**Definition 2.2.** Let be an Abelian group and *N* a commutation factor. The (complex) graded
vector space is called a color Lie (super)algebra if the following hold:

(1) g0 is a (complex) Lie algebra.

(2) For all , is a representation of . If, thendenotes the action of *X* on *Y* .

(3) For all , there exists a -equivariant map such that for all the constraint is satisfied.

(4) For all , the following Jacobi identities hold:

**Remark 2.3.** The Jacobi identity above can be rewritten in equivalent form as

Further, property (1) in Definition 2.1 is a consequence of (3) in Definition 2.2, while property (2) in Definition 2.1 is a consequence of the Jacobi identity (4) in Definition 2.2. For the particular case , reduces to a Lie algebra. If , we obtain the grading.

If, in addition, holds,is just a Lie superalgebra. Therefore, the latter condition
in Definition 2.2 points out to which extent color algebras extend ordinary Lie algebras and
superalgebras. Furthermore, if *N*(a, b) = 1 for all , is a Lie algebra graded by the group . From now Lie algebras with this last property will be called -graded Lie algebras.

The grading group inherits naturally a -grading:, where [49]. If, is called a color Lie algebra (resp., superalgebra).
Starting from a color Lie superalgebra, we define, where is the degree of *a* with respect to the -grading. It is not dicult to check that is also
a commutation factor. Furthermore, if we decompose with respect to this -grading, and introduce the Grassmann algebra, the analogue of the Grassmann-hull in the case of
Lie superalgebras, we can endow the color Lie superalgebra with a color Lie algebra structure.
Indeed, if we set , are of degree *i*, then

is a color Lie algebra with commutation factor .

**Remark 2.4.** To any associative -graded algebras with multiplication law μ, one
can associate a color Lie (super)algebra with commutation factor *N* denoted by by means of

On can easily see that the Jacobi identities are a consequence of the associativity of the product μ.

Introducing a graded basis of , *a* ∈ , the commutator is
expressed as

(2.2)

The scalars are called the structure constants of over the given basis.

**Definition 2.5.** A representation of a color Lie (super)algebra is a mapping , where is a graded vector space such that

for all .

We observe that for all *a*, *b*∈ we have , which implies that any *Va* has the
structure of a -module. Fixing an elementand denoting by its components,
we can introduce the mapping defined by , from which we conclude that. The mapping gr is called the grading map. Now, for a given
matrix representation , where, the nonvanishing indices of the matrix are thosesatisfying the equality .

**Example 2.6.** Let , and be an Abelian group of
order *n*. Let gr be defined by

Consider a commutation factor *N* satisfying the previous relations (2.1). We construct the color
algebra (Green and Jarvis in [25]) by means of its dening relations.
A basis of is given by the *m* × *m* complex matrices , with. The space is endowed with a color Lie (super)algebra structure:

Several subalgebras of can also be defined using this procedure (see [25,49]). In particular, if is a color Lie (super)algebra with basis satisfying , can be embedded into some if we define where the satisfy.

**Example 2.7. **Let be an arbitrary Lie algebra and let be a basis
of . Consider. the complex algebra generated by *e*_{1}, *e*_{2} such that

This structure, called the generalised Clifford algebra, has been studied by several authors (see [37,38,39,40] and the references cited therein). Introduce a basis of . It is easy to see that we have , and thus is a color Lie algebra for which the Abelian group is and the commutation factor is. Indeed, if we set , we have

It is a matter of a simple calculation to check the Jacobi identities. Furthermore, it is known that admits a unique irreducible *n* × *n* faithful matrix representation:

If we introduce now the *n*^{2} × *n ^{2}* matrices

and define , together with a *d*-dimensional matrix representation of given by, we obtain an *n*^{2} *d*-dimensional representation of .

This construction can be extended for larger Abelian groups. Indeed, starting from the generalised Clifford algebra generated by satisfying, and defining, we obtain in the same way a color Lie algebra with Abelian group . Finally, let us mention that a similar construction can also be obtained in straightforward way by starting from a Lie superalgebra.

A somewhat different ansatz, which turns out to be of wide interest in applications, refers to
the realization of color Lie (super)algebra in terms of differential operators [25]. Let g be a color
Lie (super)algebra with basis and commutation relations . Denote by *N* the commutation factor. Assume further that we have a *d*-dimensional
matrix representation , and introduce *d* variables θ* ^{i}* together with their associated
differential operators . As before, we assume that the index

(2.3)

with , A very elegant construction of θ* ^{i}* and can be found in [27] in terms of usual
bosons or fermions . If we set and suppose that

(2.4)

This means that the variables θ* ^{i}* are in the fundamental representation of , while the variables belong to the corresponding dual representation. These two sets of variables, generalizing
the usual bosonic and fermionic algebras, play a central role in differential realizations of . The next result shows that to a color Lie algebra we can associate a graded Lie algebra with
the same grading group . For this reason, we call it decoloration theorem.

**Theorem 2.8.** There is an isomorphism between color Lie (super)algebras and graded Lie
(super)algebras.

**Proof.** Consider a color Lie (super)algebra with
commutation factor *N* and grading group . We also introduce the commutation factor as
defined previously. In the case where is a color Lie algebra, we have = *N*. Consider now a
graded algebra with and multiplication law given by

(2.5)

such that the following constraint is satisfied [49]:

(2.6)

If we suppose that holds, then we have the equality. This implies that *G* is a subalgebra of the associative algebra defined by
equations (2.3). As a consequence, condition (2.6) is equivalent to assume the associativity of the
product in *G*. If we further suppose that is a commutation factor, the additional condition

(2.7)

is satisfied. We call σ a multiplier. In fact, it can be easily shown that equation (2.7) is a consequence of (2.6). Let us define

We observe that all elements in (for any a ∈ ) are of degree zero. For we set . From this, we derive the commutators

(2.8)

These new brackets (2.8) satisfy the Jacobi identity (for

if σ satisfies the following condition [49]:

is invariant under cyclic permutation, (2.9)

It turns out that (2.7) and (2.9) are equivalent to (2.6). This means that the algebra inherits the structure of a -graded Lie (super)algebra.

In [49], a more general result was established, and a close relationship between -graded Lie (super)algebras corresponding to different multiplication factors was established. In fact, we can even (composing the Grassmann-hull and the results of Theorem 2.8) associate a -graded-Lie algebra to a color Lie (super)algebra. This decoloration theorem was established in [1]. Let us briefly recall the main steps of its proof.

Consider a color Lie (super)algebra with commutation factor *N*. Introduce also a -graded algebra, canonically generated by the variables satisfying

(2.10)

Then the zero-graded part of ,

is a Lie algebra. Indeed, for and , it is not difficult to check that following relations are satisfied:

(2.11)

This decoloration theorem has an interesting consequence. Specifically, it means that one can associate a group to a color Lie (super)algebra and that the parameters of the transformation are related to the algebra Λ above. This result was used in the papers of Wills-Toro et al. in the trefoil symmetry frame [28,29,30,31,32,33]. Finally, let us mention that this decoloration theorem is in some sense the inverse procedure to the one given in Example 2.7.

Lie algebras of order *F*, introduced in [22,23,24], correspond to a different kind of extensions of
Lie (super)algebras, motivated by the implementation of nontrivial extensions of the Poincare
algebra in QFT. This type of algebras is characterized by a hybrid multiplication law: part of
the algebra is realized by a binary multiplication, while another part of the algebra is realized
via an *F*-order product. More precisely, a Lie algebra of order *F* is graded by the Abelian group . The zero-graded part is a Lie algebra and an *F*-fold symmetric product (playing the
role of the anticommutator in the case *F* = 2) expresses the zero graded part in terms of the
nonzero graded part.

**Definition 3.1.** Let -graded -vector space is called a complex Lie algebra of order *F* if the following hold:

(1) is a complex Lie algebra.
(2) For all *i* = 1, . . . , *F* − 1, is a representation of . If , then [*X*, *Y* ] denotes the action of *X* on *Y* for any *i* = 1, . . . , *F* − 1.
(3) For all *i* = 1, . . . , *F* − 1, there exists an F-linear, -equivariant map , where denotes the *F*-fold symmetric product of .
(4) For all and , the following "Jacobi identities" hold:

(3.1)

**Remark 3.2.** If *F* = 1, by definition = and a Lie algebra of order 1 is a Lie algebra. If
*F* = 2, then g is a Lie superalgebra. Therefore, Lie algebras of order *F* appear as some kind of
generalizations of Lie algebras and superalgebras.

**Proposition 3.3.** Let be a Lie algebra of order *F*, with *F* > 1. For any *i* = 1, . . . , *F* − 1, the -graded vector spaces is a Lie algebra of order *F*. We call
these type of algebras elementary Lie algebras of order *F*.

**Remark 3.4. **Let be an associative -graded algebra with multiplication
μ. One can associate a Lie algebra of order *F* to *A* as follows. For any , *i* = 1, . . . , *F* − 1, we have

Furthermore, one can easily see that the Jacobi identities are a consequence of the associativity
of the product μ. Moreover, if *A* is an associative algebra and the commutative *F*-dimensional
algebra generated by a primitive element *e* such that *e*^{F} = 1, the algebra is -graded and thus leads to a Lie algebra of order *F*.

**Definition 3.5.** A representation of an elementary Lie algebra of order *F* is a linear map , such that for all,

(3.2)

being the symmetric group of *F* elements.

By construction, the vector space *V* is graded , and for all *a* = {0, . . . , *F*−1}, V_{a} is a -module. Further, the condition holds.

**Theorem 3.6** (see [22,23,24]). Let be a Lie algebra and be a -module such that

(i) is a Lie algebra of order F_{1} > 1;
(ii) admits a -equivariant symmetric form of order F_{2} > 1.

Then inherits the structure of a Lie algebra of order F_{1} + F_{2}.

The theorem above can be generalized to include the case F_{1} = 1 [22,23,24].

**Example 3.7 **(this is a consequence of Theorem 3.6, modied to include F_{1} = 1). Let be any
Lie algebra and let be its adjoint representation. Introduce a basis of , the corresponding basis of , and the Killing form.
Then one can endow with a Lie algebra of order three structure given by

**Example 3.8.** Let be the Poincare algebra in
*D*-dimensions and let be the *D*-dimensional vector representation
of . The brackets

with the metric endow with an elementary Lie algebra of order three structure which is denoted by .

**Example 3.9.** Let mat(*m*_{1}, *m*_{2}, *m*_{3}) and mat_{el}(*m*_{1}, *m*_{2}, *m*_{3}) be the set of (*m*_{1} + *m*_{2} + *m*_{3}) ×
(*m*_{1} + *m*_{2} + *m*_{3}) matrices of the form

(3.3)

with , . basis of this set
of matrices can be constructed as follows. Consider the (*m*_{1} + *m*_{2} + *m*_{3})^{2} canonical matrices. With the following convention for the indices, , the generators are given by

Writing and , we denote generically by the canonical generators of degree zero, by those of degree one, and by those of degree two. With these conventions, the brackets read

(3.4)

This shows that is endowed with the structure of
Lie algebra of order three (resp., a structure of an elementary Lie algebra of order three). In
particular, when (*m*_{1}, *m*_{2}, *m*_{3}) (resp., mat_{el}(*m*_{1}, *m*_{2}, *m*_{3})) is endowed with the structure of
Lie algebra of order three (resp., a structure of an elementary Lie algebra of order three). In
particular, when *m*_{1} = *m*_{2} = *m*_{3}, the algebra above can be rewritten as mat(*m*, *m*, *m*) = being a faithful matrix representation of the canonical generator of.

The question to nd appropriate variables to represent Lie algebras of order *F* is much more
involved than for color algebras. However, in some specic cases, we were able to nd appropriate
variables (see [34,35,36]), and it turns out that these variables are strongly related to Clifford
algebras of polynomials [44,45,46,47,48]. We will give another realization below.

Color Lie (super)algebras of order *F* can be seen as a synthesis of the two types of algebras introduced
previously. Indeed, for such algebras, we have simultaneously a binary product associated
with a commutation factor and an *F*-order product. The latter is no more fully symmetric, but is
also associated with the commutation factor. In this section, we focus on color Lie (super)algebra
of order three.

**Definition 4.1.** Let be an Abelian group and let *N* be a commutation factor, is
an elementary color Lie (super)algebra of order three if the following hold:

(1) is a color Lie (super)algebra.

(2) is a representation of . If are homogeneous elements, then denotes the action of X on Y .

(3) There exists a -equivariant map such that for all we have

(4) The following "Jacobi identities" hold:

We observe that if is a decomposition of with respect to its-grading, as seen in Section 2, and such that , then is called a color Lie algebra (resp., superalgebra). Moreover, if and hold, the algebra g is called a Lie superalgebra of order three.

**Definition 4.2.** A representation of an elementary color Lie (super)algebra of order three is
a linear map satisfying the conditions

By construction, the vector space *V* is graded and we have . Furthermore, each *V _{i,a} *is a g

**Remark 4.3.** Let

be an associative -graded algebra with multiplication μ. One can associate a color Lie
superalgebra of order three to *A* defining the products in a similar manner as in Remarks 2.4 and 3.4. In this case, the Jacobi identities are also a consequence of the associativity of the product μ. Similarly, if *A* is an associative -algebra and the commutative three-dimensional algebra generated by a primitive element e such that *e*^{3} = 1, the algebra is associative and -graded, and therefore leads to a color Lie algebra
of order three.

The examples of color Lie (super)algebras of order *F* are basically of two types: we can
construct a color Lie (super)algebra of order *F* from either a color Lie (super)algebra or a Lie
algebra of order *F*.

**Example 4.4.** Let be the color Lie (super)algebra of Example 2.6 and
let be the generalised Clifford algebra with canonical generator *e*, then

(1) is a color Lie (super)algebra of order three;

(2) is an elementary color Lie (super)algebra of order three.

For the second algebra, following the notations of Example 2.6, we denote by a basis of , and a basis ofof . Then, the trilinear brackets read

**Example 4.5.** Let be an arbitrary (elementary) Lie algebra of order three and let be the
generalized Clifford algebra with canonical generators e_{1}, e_{2}. Then is a color Lie algebra of
order three with Abelian group and commutation factor . Suppose that an elementary Lie algebra of order three is given. Denote by a basis of (resp., ) such that

Define and ,

we thus have that

is an elementary color Lie algebra of order three with brackets

As in Example 2.7, this can be extended for and for color Lie superalgebras of order three.

Example 4.6. This example is a synthesis of Examples 2.6 and 3.9. Consider three Abelian
groups and corresponding commutation factors *N*_{1}, *N*_{2}, *N*_{3}. Then we dene on the
group the commutation factor, with etc. Let with *i* = 1, 2, 3 be three integers and let be three color Lie (super)algebras as in Example 2.6.
Introduce now the matrices in the fundamental representation of and in the dual of the fundamental representation of. In a similar way as in
Example 3.9, we consider the set of matrices and the algebra

with the notations of Example 3.9. It is obviously a color Lie (super)algebra of order three. The various brackets are similar to those of Example 3.9 and 2.6. We just give a few brackets for completeness:

To conclude this section, we now show that there is an analogous of the decoloration theorem
established in Section 2. As done there, one can proceed in two different (but related) ways.
To set up the main result of this theorem, consider a color Lie
superalgebra with grading Abelian group and commutation factor *N*. In the second approach,
we directly associate to a Lie algebra of order three, in the same manner as in Section 2
by considering the algebra , where Λ_{a} is generated by the variables θ^{a}_{i} satisfying equation (2.10). The algebra

is a Lie algebra of order three. This is proved in a similar way as in Section 2 and only the trilinear brackets are slightly different. Let . It is not dicult to check that

The Jacobi identities involving trilinear brackets are a consequence of the identity (for any, *i* = 1, . . . , 4) together with the associativity of the product in Λ and equation (2.10). This proves that is a Lie algebra of order three.

In the first correspondence, we introduce as in Section 2 and the variables *e*_{a} as in
Theorem 2.8 satisfying equations (2.5) and (2.6). Recall that the last property ensures that
the product is associative. Then the algebra is a Lie
(super)algebra of order three. The proof goes along the same lines as in Theorem 2.8. For the
bilinear part, the proof is the same as in Theorem 2.8. For the cubic bracket, if we take , , and a simple calculation shows, using condition (2.6), the explicit structure of
the trilinear bracket:

where denotes the degree of *X* with respect to the grading of etc. Since we
have , and the algebra *G* is associative, there is no need to prove the
Jacobi identities involving trilinear brackets (the proof being the same as in previous cases). This
illustrates how we can associate a Lie (super)algebra of order three to a color Lie (super)algebra
of order three. These results, taken together, can be resumed in uniform manner in the following
decoloration theorem.

**Theorem 4.7.** There is an isomorphism between color Lie (super)algebras of order three and
Lie (super)algebras of order three.

To finish this section, let us observe the following. The decoloration theorem above and that
of Section 2 seem to indicate that color Lie (super)algebras (resp., color Lie (super)algebras
of order three) do not really constitute new objects, since they are isomorphic to Lie algebras
(resp., Lie algebras of order three). In fact, as a consequence of these theorems, for any representation
*R* of a color algebra , we can construct, by means of the procedure above, an isomorphic
representation of the associated noncolor algebra. The converse of this procedure also holds. It
should, however, be taken into account that this property does not imply that all representations
of color (resp., noncolor) algebras are obtained from representations of the corresponding
noncolor (resp., color) algebras.^{1}

Quons were conceived in particle statistics as one of the alternatives to construct theories where either the Bose or Fermi statistics are violated by a small amount [20]. Although observables related to particles subjected to this type of intermediate statistics fail to have the usual locality properties, their validity in nonrelativistic field theory and free field theories obeying the TCP theorem has been shown. In this section, we prove that color Lie algebras of order three admitting a nite-dimensional linear representation can be realized by means of quon algebras for the important case q = 0. This result is a generalization of various properties that are well known for the usual boson and fermion algebras.

Let −1 ≤ q ≤ 1 and consider the variables a_{i}; ai, i = 1, . . . , n. We dene the q-mutator or
quon algebra by means of

(5.1)

where no relations between variables of the same type are postulated. The (complex) quon
algebra is denoted by For the two extreme values of q, we recover the well-known
statistics. If q = −1, together with the relations a^{i}a^{j} + a^{j}a^{i} = 0 and a_{i}a_{j} + a_{j}a_{i} = 0, the
quon algebra reduces to the fermion algebra. For q = 1, together with a^{i}a^{j} − a^{j}a^{i} = 0 and
a_{i}a_{j} − a_{j}a_{i} = 0, it reproduces the boson algebra. Therefore, the quon algebra can be interpreted
as an interpolation between Bose and Fermi statistics.^{3}

**Lemma 5.1.** Let *M*_{1}, . . . , *M*_{k} be (*n* × *n*) complex matrices satisfying a polynomial relation
P(*M*_{1}, . . . , *M*_{k}) = 0. Then there exists *k* elements (*k* = 1, . . . ,*n*) such that
P(*M*_{1}, . . . , *M*_{n}) = 0.

**Proof.** Given two arbitrary generators , by equation (5.1) we have a_{i}a^{j} = . This means in particular that the n^{2} elements e^{i}_{j} defined by e^{i}_{j} = a^{i}a_{j} , 1 ≤ i, j ≤ n, of satisfy the relation. Denoting by E^{i}_{j} the canonical generators of (the (*n* × *n*) complex matrices), the mapping defined by f(E^{i}_{j}) = e^{i}_{j} turns out
to be a injection. Therefore, since there is no kernel, the elements have to satisfy the same relations as the matrices M_{k}. Thus P(*M*_{1}, . . . , *M*_{n}) = 0.

The quon algebra with q = 0 has been studied in detail by Greenberg, and constitutes an example of "infinite statistics" [20]. It was moreover shown there that the q = 0 operators can be used as building blocks for representations in the general case. We next show that, under special circumstances, color Lie algebras of order three naturally embed into a q = 0 quon algebra.

**Theorem 5.2.** Let be an Abelian group, *N* a commutation factor, and a color Lie
(super)algebra of order three. If admits a nite-dimensional matrix representation, then can be realized by a quon algebra with q = 0.

**Proof.** Suppose that the decomposition with respect to the
Abelian group is given. Let a basis of and be a basis of _{1,a} such that the
following relations hold:

(5.2)

Let ρ be an *n*-dimensional representation of and let denote
the corresponding transformed basic elements. Then the representation space V on which the
matrices M and N act satises the decomposition . Now, since
the inclusions are satisfied, we can nd a basis of V such
that , i.e., with respect to the grading group , the block is
of degree *i*, for *i* = 0, 1, 2. With respect to this basis, the matrices *M* and *N* can be rewritten as

Let n_{i} = dim V_{i}, *i* = 1, 2, 3, where obviously n_{1} + n_{2} + n_{3} = n. We denote by the components of the vector, and the matrix elements of M and . From
now on, we adopt the convention that an index in the form i_{a}, a = 0, 1, 2 is of degree *a* with
respect to the grading group . Furthermore, using the same notations as in Section 2 with
respect to the grading group is of degree gr(i_{a}). This in particular implies some relations for
the matrix elements of M and N. For instance, considering the matrix element , we have a = gr(i_{2})− gr(j_{1}), and so forth. Consider now three series of quons , such
that for any m ≠ n, the relation a_{mi}a_{n} ^{j} = 0 holds. It follows from the grading group that
a_{a}^{ia} (resp., ) is of degree a (resp., of degree −a), while, with respect to the group ,a_{a}^{ia} (resp., ) is of degree gr(i_{a}) (resp., − gr(i_{a})). We now define

By denition, the matrices and N_{m}^{ (a)} satisfy the relations (5.2). Now, applying
Lemma 5.1, the elements satisfy the same
relations. Furthermore, since the quon algebra is an associative algebra, the Jacobi identities
are automatically satisfied. Therefore, the color algebra g has been realized in the quon algebra , finishing the proof.

It should be observed that certain types of Lie algebras of order three do not admit nitedimensional matrix representations. However, these can realized by means of Clifford algebras of polynomials [34,35,36,44,45,46,47,48]. Moreover, a similar argumentation allows to realize any given type of algebra admitting nite-dimensional representations by an appropriate set of quons with q = 0.

During the preparation of this work, R. Campoamor-Stursberg was nancially supported by the research projects MTM2006-09152 (M.E.C.) and CCG07-UCM/ESP-2922 (U.C.M.-C.A.M.).

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