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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Color Lie algebras and Lie algebras of order F

R. CAMPOAMOR-STURSBERG1 and M. RAUSCH DE TRAUBENBERG2

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

2Laboratoire 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

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Abstract

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 Cli ord-like algebras. It is, moreover, shown that color algebras admit realizations as q = 0 quon algebras.

1 Introduction

The problem of nding mass formulae for particles belonging to a representation of an interaction group motivated, in the beginning sixties, the e orts 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 Image and the orthosymplectic algebraImage 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 Image[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 Image, where C is a Clifford algebra C and D aImage -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 uni ed 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 uni cation 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.

2 Color Lie algebras

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 de ne color Lie (super)algebras is a grading determined by an Abelian group. The latter, besides de ning the underlying grading in the structure, moreover, provides a new object known as commutation factor.

Definition 2.1. LetImagebe an Abelian group. A commutation factor N is a map ImageImagesatisfying the following constraints:

Image

Image

Image

The de nition above implies, in particular, the following relations:

Image (2.1)

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

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

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

(2) For all Image, Image is a representation of Image. IfImage, thenImagedenotes the action of X on Y .

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

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

Image

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

Image

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 Image, Image reduces to a Lie algebra. If Image, we obtain the gradingImage.

If, in addition, Image holds,Imageis 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 Image, Imageis a Lie algebra graded by the group Image. From now Lie algebras with this last property will be called Image-graded Lie algebras.

The grading group Image inherits naturally a Image-grading:Image, where ImageImage [49]. IfImage, Image is called a color Lie algebra (resp., superalgebra). Starting from a color Lie superalgebra, we defineImage, whereImage is the degree of a with respect to the Image-grading. It is not dicult to check that Image is also a commutation factor. Furthermore, if we decompose Image with respect to this Image-grading, and introduce the Grassmann algebraImage, 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 Image, are of degree i, then

Image

is a color Lie algebra with commutation factor Image.

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

Image

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

Introducing a graded basis Image of Image, aImage, the commutator is expressed as

Image (2.2)

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

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

Image

for all Image.

We observe that for all a, bImage we haveImage , which implies that any Va has the structure of a Image-module. Fixing an elementImageand denoting by Image its components, we can introduce the mapping Image defined by Image, from which we conclude thatImage. The mapping gr is called the grading map. Now, for a given matrix representation Image, whereImage, the nonvanishing indices of the matrixImage are thoseImagesatisfying the equality Image.

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

Image

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

Image

Several subalgebras of Image can also be defined using this procedure (see [25,49]). In particular, if Image is a color Lie (super)algebra with basisImage satisfying Image, Image can be embedded into someImage if we defineImage where theImage satisfyImage.

Example 2.7. Let Image be an arbitrary Lie algebra and letImage be a basis of Image. ConsiderImage. the complex algebra generated by e1, e2 such that

Image

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

Image

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

Image

If we introduce now the n2 × n2 matrices

Image

and define Image, together with a d-dimensional matrix representation of Image given byImage, we obtain an n2 d-dimensional representation of Image .

This construction can be extended for larger Abelian groups. Indeed, starting from the generalised Clifford algebra Image generated byImage satisfyingImage, Image and definingImage, we obtain in the same way a color Lie algebra with Abelian group Image. 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 Image and commutation relationsImageImage . Denote by N the commutation factor. Assume further that we have a d-dimensional matrix representation Image, and introduce d variables θi together with their associated differential operators Image. As before, we assume that the index i is of degreeImage is of degreeImage , and Image of degree gr(i) subjected to the following commutation relations:

Image (2.3)

with Image, A very elegant construction of θi and Image can be found in [27] in terms of usual bosons Image or fermionsImage . If we setImage and suppose that gr(a) = gr(j) − gr(i), a direct computation gives

Image (2.4)

This means that the variables θi are in the fundamental representation of Image , while the variables Image 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 Image. The next result shows that to a color Lie algebra we can associate a graded Lie algebra with the same grading group Image. 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 Image with commutation factor N and grading group Image. We also introduce the commutation factor Image as defined previously. In the case where Imageis a color Lie algebra, we have Image = N. Consider now a graded algebra Image with Image and multiplication law given by

Image (2.5)

such that the following constraint is satisfied [49]:

Image (2.6)

If we suppose that Image holds, then we have the equalityImageImage. 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 Image is a commutation factor, the additional condition

Image (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

Image

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

Image (2.8)

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

Image

if σ satisfies the following condition [49]:

Image is invariant under cyclic permutation, Image (2.9)

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

In [49], a more general result was established, and a close relationship between Image -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 Image-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 Image a color Lie (super)algebra with commutation factor N. Introduce also Image a Image-graded algebra, canonically generated by the variables Image satisfying

Image(2.10)

Then the zero-graded part of Image,

Image

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

Image (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.

3 Lie algebras of order F

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 Image. 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 Image-graded Image-vector space Image is called a complex Lie algebra of order F if the following hold:

(1) Image is a complex Lie algebra. (2) For all i = 1, . . . , F − 1, Image is a representation of Image. If Image, 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, Image-equivariant map Image, whereImage denotes the F-fold symmetric product of Image. (4) For all Image and Image , the following "Jacobi identities" hold:

Image (3.1)

Remark 3.2. If F = 1, by definition Image= Image 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 Image be a Lie algebra of order F, with F > 1. For any i = 1, . . . , F − 1, the Image-graded vector spaces Image is a Lie algebra of order F. We call these type of algebras elementary Lie algebras of order F.

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

Image

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 Image the commutative F-dimensional algebra generated by a primitive element e such that eF = 1, the algebraImageImage is Image-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 Image, such that for allImage,

Image (3.2)

Imagebeing the symmetric group of F elements.

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

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

(i) Image is a Lie algebra of order F1 > 1; (ii) Image admits a Image-equivariant symmetric form of order F2 > 1.

Then Image inherits the structure of a Lie algebra of order F1 + F2.

The theorem above can be generalized to include the case F1 = 1 [22,23,24].

Example 3.7 (this is a consequence of Theorem 3.6, modi ed to include F1 = 1). Let Image be any Lie algebra and let Image be its adjoint representation. IntroduceImage a basis of Image, Image the corresponding basis of Image, and Image the Killing form. Then one can endow Image with a Lie algebra of order three structure given by

Image

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

Image

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

Example 3.9. Let mat(m1, m2, m3) and matel(m1, m2, m3) be the set of (m1 + m2 + m3) × (m1 + m2 + m3) matrices of the form

Image (3.3)

with Image, ImageImage. basis of this set of matrices can be constructed as follows. Consider the (m1 + m2 + m3)2 canonical matricesImage. With the following convention for the indicesImage, Image, the generators are given by

Image

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

Image (3.4)

This shows that Image 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 (m1, m2, m3) (resp., matel(m1, m2, m3)) 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 m1 = m2 = m3, the algebra above can be rewritten as mat(m, m, m) = Imagebeing a faithful matrix representation of the canonical generator ofImage.

The question to nd appropriate variables to represent Lie algebras of order F is much more involved than for color algebras. However, in some speci c 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.

4 Color Lie algebras of order F

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 Imagebe an Abelian group and let N be a commutation factor, Image is an elementary color Lie (super)algebra of order three if the following hold:

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

(2) Image is a representation of Image. IfImage are homogeneous elements, thenImage denotes the action of X on Y .

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

Image

(4) The following "Jacobi identities" hold:

Image

We observe that if Image is a decomposition of Image with respect to itsImage-grading, as seen in Section 2, and such that Image, then Image is called a color Lie algebra (resp., superalgebra). Moreover, if Image and Image 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 Image satisfying the conditions

Image

By construction, the vector space V is graded and we have ImageImage. Furthermore, each Vi,a is a g0,0-module and the inclusion relation Image holds.

Remark 4.3. Let

Image

be an associative Image-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 Image-algebra and Image the commutative three-dimensional algebra generated by a primitive element e such that e3 = 1, the algebra ImageImage is associative and Image-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 Image be the color Lie (super)algebra of Example 2.6 and let Image be the generalised Clifford algebra with canonical generator e, then

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

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

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

Image

Example 4.5. Let Image be an arbitrary (elementary) Lie algebra of order three and letImage be the generalized Clifford algebra with canonical generators e1, e2. ThenImage is a color Lie algebra of order three with Abelian group Image and commutation factorImage . Suppose that an elementary Lie algebra of order three Image is given. Denote byImageImage a basis of Image (resp., Image) such that

Image

Define Image and Image,

we thus have that

Image

is an elementary color Lie algebra of order three with brackets

Image

As in Example 2.7, this can be extended for Image 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 Image and corresponding commutation factors N1, N2, N3. Then we de ne on the group Image the commutation factorImage, withImage etc. LetImage with i = 1, 2, 3 be three integers and let Image be three color Lie (super)algebras as in Example 2.6. Introduce now the matrices Image in the fundamental representation of Image and in the dual of the fundamental representation ofImage. In a similar way as in Example 3.9, we consider the set of matrices Image and the algebra

Image

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:

Image

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 Image a color Lie superalgebra with grading Abelian group Imageand commutation factor N. In the second approach, we directly associate to Imagea Lie algebra of order three, in the same manner as in Section 2 by considering the algebra Image, where Λa is generated by the variables θai satisfying equation (2.10). The algebra

Image

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 ImageImage. It is not dicult to check that

Image

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

In the first correspondence, we introduce Image as in Section 2 and the variables ea 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 Image 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 Image, Image, and Image a simple calculation shows, using condition (2.6), the explicit structure of the trilinear bracket:

Image

where Image denotes the degree of X with respect to the Image grading of Image etc. Since we have Image, 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 Image, 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

5 Quons and realization of color Lie (super)algebras of order three

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 ai; ai, i = 1, . . . , n. We de ne the q-mutator or quon algebra by means of

Image (5.1)

where no relations between variables of the same type are postulated. The (complex) quon algebra is denoted by ImageFor the two extreme values of q, we recover the well-known statistics. If q = −1, together with the relations aiaj + ajai = 0 and aiaj + ajai = 0, the quon algebra reduces to the fermion algebra. For q = 1, together with aiaj − ajai = 0 and aiaj − ajai = 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 M1, . . . , Mk be (n × n) complex matrices satisfying a polynomial relation P(M1, . . . , Mk) = 0. Then there exists k elements Image(k = 1, . . . ,n) such that P(M1, . . . , Mn) = 0.

Proof. Given two arbitrary generators Image, by equation (5.1) we have aiaj = Image. This means in particular that the n2 elements eij defined by eij = aiaj , 1 ≤ i, j ≤ n, of Image satisfy the relationImage. Denoting by Eij the canonical generators of Image (the (n × n) complex matrices), the mappingImage defined by f(Eij) = eij turns out to be a injection. Therefore, since there is no kernel, the elements Image have to satisfy the same relations as the matrices Mk. Thus P(M1, . . . , Mn) = 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 Image 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 Image be an Abelian group, N a commutation factor, and Imagea color Lie (super)algebra of order three. If Image admits a nite-dimensional matrix representation, then Image can be realized by a quon algebra with q = 0.

Proof. Suppose that the decomposition Image with respect to the Abelian group Imageis given. Let Imagea basis of Image andImage be a basis of Image1,a such that the following relations hold:

Image (5.2)

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

Image

Let ni = dim Vi, i = 1, 2, 3, where obviously n1 + n2 + n3 = n. We denote by Image the components of the vectorImage, and the matrix elements of M and Image. From now on, we adopt the convention that an index in the form ia, a = 0, 1, 2 is of degree a with respect to the grading group Image. Furthermore, using the same notations as in Section 2 with respect to the grading group Image is of degree gr(ia). This in particular implies some relations for the matrix elements of M and N. For instance, considering the matrix element Image, we have a = gr(i2)− gr(j1), and so forth. Consider now three series of quons Image, Image such that for any m ≠ n, the relation amian j = 0 holds. It follows from the grading group Image that aaia (resp., Image) is of degree a (resp., of degree −a), while, with respect to the group Image,aaia (resp., Image) is of degree gr(ia) (resp., − gr(ia)). We now define

Image

By de nition, the matrices Image and Nm (a) satisfy the relations (5.2). Now, applying Lemma 5.1, the elements Image 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 Image, 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.

Acknowledgments

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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