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Classification of Finite Spectral Triples

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings.


Introduction
It is nowadays admitted that the major difficulty in quantizing gravity lies in our current conception of geometrical spaces at very short scale. Therefore, a first step towards a quantum theory of gravitation may be a suitable redefinition of the geometrical notions. Due to Heisenberg's uncertainty relations, in a quantum theory, the ordinary notion of a point disappears, so that we have to develop a geometry without any reference to points. This is the basic principle of noncommutative geometry, where points are replaced by the algebra of coordinates, which may be noncommutative. There are many possible choices of the algebra of coordinates. For example, we can replace the ordinary plane, whose coordinates satisfy xy = yx, by the quantum plane, i.e. the algebra generated by the relation xy = qyx for a complex number q. This leads us naturally, when we aim at describing the symmetries of the quantum plane, to the theory of quantum groups.
Another possible choice of the algebra of coordinates lies in replacing it by a suitable involutive subalgebra of the algebra of operators on a Hilbert space. Then, we try to extend, guided by the analogy with quantum mechanics, the ordinary definitions and theorems of differential geometry, to the noncommutative case. This is the framework of noncommutative geometry, a theory fully developped in [1]- [6]. Since matter is fermionic, the notion that should be extended to noncommutative geometry if that of a spin structure on a riemannian manifold. This has been achieved via spectral triples in [4], [7]. By definition, a spectral triple is a triple (A, H, D), where A is a real involutive algebra, that stands for the algebra of coordinates, represented on the fermionic Hilbert space H. D is a self-adjoint operator acting on H, which is the generalisation of the Dirac operator. To provide a suitable extension of spin geometry, these three objects are assumed to fulfill some axioms called axioms of noncommutative geometry.
Starting with a suitable class of spectral triples, we can construct gauge theories with spontaneous symmetry breaking. In that case, both the gauge and the Higgs fields appear as connections, the former on an ordinary bundle, the latter on a finite one. Accordingly, the spectral triples we use are products of two spectral triples. The first one encodes all data concerning the geometry of spacetime and the second one is what we call a finite spectral triple.
In section 2, we define finite spectral triples and give the relation between the general axioms of noncommutative geometry and the simplest case of finite noncommutative geometry. Section 3 is devoted to a general resolution of the equations we obtained. In section 4, we introduce some diagrams that allow to classify and to construct finite spectral triples. Finally, in section 5 we give some applications to model building in particle physics and determine the general feature of the corresponding Yang-Mills-Theory within the diagrammatic approach.

Axioms for finite spectral triples
Finite spectral triples are particular cases of spectral triples of dimension 0. The latter are rigorously defined within the axioms of noncommutative geometry and yield a general theory of discrete spaces. Among all discrete spaces, we focus on finite ones, thus the algebra is finite dimensional. Futhermore, we will also assume that the Hilbert space is finite dimensional, an infinite dimensional one corresponding to a theory with an infinite number of elementary fermions. Accordingly, a finite spectral triple (A, H, D) is defined as a spectral triple of dimension 0 such that both A and H are finite dimensional. Using such a triple, it is possible to a construct Yang-Mills theory with spontaneous symmetry breaking whose gauge group is the group of unitary elements of A, H is the fermionic Hilbert space and D is the mass matrix.
It is known that finite dimensional real involutive algebras which admit a faithful representation on a finite dimensional Hilbert space are just direct sums of matrix algeras over the fields of real numbers, complex numbers and quaternions. Therefore, we write the algebra as a direct sum where M n (K) denotes the algebra of square matrices of order n with entries in the field K = R, C or H. This remark simplifies considerably finite noncommutative geometry and it becomes possible to give a detailed account of all finite spectral triples. We also consider the particular case of complex finite spectral triples, whose algebra is a complex algebra and the representation π is assumed to be linear over complex numbers.
In this section, we give the axioms for finite spectral triples and sketch their relations with the general case. Since both the algebra A and the Hilbert space H are finite dimensional, all assumptions concerning functional analysis, that are necessary in the general case, are obviously satisfied. Thus, there is a overwhelming simplification in the formulation of noncommutative geometry and the seven axioms involved in [7] reduce to the four following ones: reality, orientability, Poincaré duality and first order condition. To proceed, we fix a finite spectral triple (A, H, D) together with a faithful and unitary representation π of A as operators on H. Moreover, the formulation of the axioms of noncommutative geometry involves two other operators, the charge conjugation J and the chirality χ that we shall also fix. The precise definition of these two operators will become clearer in the following sections.

Reality
To define noncommutative manifolds, we need, in this setting, a suitable notion of Poincaré duality. A first step towards a formulation of Poincaré duality in noncommutative geometry lies in introducing a bimodule structure on H. Note that the bimodule structure is also necessary if one wants to incorporate the color sector in the Connes-Lott model [3]. However, this bimodule is not arbitrary and it has to fulfill the reality axiom.
Then we can equip H with a bimodule structure. The left action is (x, ψ) ∈ A × H → xψ = π(x)ψ, and the right action is This defines an (A, A)-bimodule structure or equivalently, a (A ⊗ A op )-module structure on H. Note that at a mathematical level, the introduction of this operator has been motivated by the transition from complex K-theory to real K-theory that turns out to be necessary in the formulation of noncommutative spin geometry. Moreover, given a cyclic and separating vector in H, J is Tomita-Takesaki involution that allows us to pass from A to its commutant A ′ . For the standard model, this operator is just charge conjugation. In other words, the Hilbert space of the finite spectral triple of the standard model is the vector space spanned by all fermions and antifermions and the operator J is the antilinear operator mapping a fermion to the corresponding antifermion. We gave the reality axiom in the finite case only. For general spectral triples, we only have to change some signs in the relations involving J , depending on the dimension modulo 8 of the spectral triple [7]. In the commutative case of a smooth manifold V endowed with a spin structure, J is the charge conjugation defined on the space of square integrable sections of the spinor bundle of V .

Orientability
Since the noncommutative manifold defined by a spectral triple has to be oriented, we must specify an orientation form. In the case of an ordinary manifold of dimension n, this is achieved by means of a nowhere vanishing n-form γ. The extension of this notion to the noncommutative case goes through the deep relation, in the commutative case, between differential forms and Hochschild homology.
Let us now briefly recall some elementary aspects of Hochschild homology, referring to [10] for a general theory. Given an associative, unital, but not necessarily commutative algebra A, and a bimodule M over A, we first define, for any integer n, the vector space T n (A, M) as the tensor product of M by n copies of A Then, we define, for any n > 0, the boundary operation b n as a linear map from T n (A, M) to For n = 0 we set b 0 = 0. By definition, the Hochschild cycles Z n (A, M) of dimension n are the elements of T n (A, M) whose image under b n is zero. Now, let us consider the bimodule M = A ⊗ A op , whose bimodule structure is given by for any y, y ′ ∈ A and x ⊗ x ′ ∈ A ⊗ A op .
These rather abstract objects are represented as operators on the Hilbert space. Using the Dirac operator D, the representation π can be extended to the Hochschild cycles as for any x ⊗ x ′ ⊗ y 1 ⊗ ... ⊗ y n ∈ Z n (A, A ⊗ A op ). This leads to a natural extension of the orientation of a manifold in the noncommutative case [7].

Orientability
For a noncommutative geometry of dimension n (n even), the chirality χ is a hermitian involution that commutes with π(x) for any x ∈ A, anticommutes with the Dirac operator D and is the image of an n-dimensional Hochschild cycle.
Before we focus on finite spectral triples, let us check that it holds for the ordinary euclidian geometry of spacetime. The latter is a compact four dimensional manifold V , the algebra A is the algebra C ∞ (V ) of complex valued smooth functions on V , the Hilbert space H is the space L 2 (V, S) of square integrable sections of the spinor bundle of V , D is the ordinary Dirac operator defined in a local chart by D = iγ µ ∂ µ , γ µ being the usual euclidian Dirac matrices, and J is the charge conjugation of Dirac spinors. The functions act simply on H by multiplication and we have, for any f ∈ C ∞ (V ), Since the algebra is commutative, the bimodule M involved in the definition of Hochschild homology is not relevant and we simply take the Hochschild cycles Z n (A, M) with M = A. Accordingly, any 0-form is a function and can be written as the image of a 0 dimensional cycle. Moreover, Hochschild cycles of dimension 1 give us 1-forms. Indeed, starting with the cycle that can be considered as the 1-form A µ dx µ if we identify i f i 0 ∂ µ f i 1 with A µ and iγ µ with dx µ . At higher orders, the situation is more subtle but we can still construct, using antisymmetrisation, some Hochschild cycles whose images are differentials forms if we identify iγ µ with dx µ . In fact, for any f 0 , f 1 , ..., f n ∈ C ∞ (V ), we define where the sum runs over all permutations of 1, 2, .., n and ǫ(σ) is the signature of σ. Due to the commutativity of C ∞ (V ), we check that c is a Hochschild cycle and that, thanks to the antisymmetrisation procedure, its image is a n-form. This applies for the volume form that can be defined as If we identify dx µ with iγ µ , the previous relation is the definition of γ 5 so that γ 5 turns out to be the chirality, and it corresponds to the volume form. Let us give two remarks pertaining to the commutative case. First, the antisymmetrisation procedure yields us immediately differential forms, so that we do not have to divide by some auxiliary fields (the so-called "junk" defined in [3]). Secondly, we have seen that Hochschild cycles provide us with differential forms when represented as operators on the Hilbert space. Thus, it is natural to ask for an exterior derivative in this framework. More precisely, we would like to have some linear maps d n from Z n (A, A) into Z n+1 (A, A) such that d n+1 • d n = 0 and such that, when represented with Dirac matrices, they coincide with the genuine exterior derivatives. In fact, we can set d n = B n , where B n is Connes' coboundary map [10]. For an associative and unital algebra A, the latter is defined as a linear map from T n (A, for any x 0 , ..., x n ∈ A. It is easy to check that B n−1 • b n + b n+1 • B n = 0. Therefore, B n (Z n (A, A)) is included in Z n+1 (A, A). Moreover, one has B n+1 • B n = 0 and thus B n can be considered as an exterior derivative in the framework of Hochschild homology. Besides, in the commutative case, when we lift this coboundary operation on operators on the Hilbert space using Dirac matrices, it yields the genuine exterior derivative without intoducing auxiliary fields. Let us check it for 1-forms. If A µ dx µ is a 1-form, it can be written as the image of the The action of B 1 on this cycle is given by When we represent it, we end up with with F µν = ∂ µ A ν − ∂ µ A ν and if we identify iγ µ with dx µ . Let us note that this also works for higher order forms. For finite spectral triples, the orientability axiom requires that χ must be written as the image of a Hochschild cycle of dimension 0. Since all elements of T 0 (A, A ⊗ A op ) are cycles, it only states that there are some elements a i , b i ∈ A such that χ = i π(a i )J π(b i )J −1 . Accordingly, the orientability axiom is reformulated as follows.

Orientability (finite case)
The chirality χ is an hermitian involution that commutes with π(x) for any x ∈ A, anticommutes with D and can be written as For the standard model, χ is simply the operator of H that takes the value −1 for left-handed fermions and +1 for right-handed fermions.

Poincaré duality
For a closed Riemannian manifold V of dimension n, there is a well known isomorphism between the de Rham groups H p (V ) and H n−p (V ) given by the bilinear map where f and g are closed p and (n-p)-forms that are representatives of the de Rham groups H p (V ) and H n−p (V ). Formulating this in noncommutative geometry is a rather difficult task that requires the use of Kasparov's biinvariant theory. We refer to [2] and [7] for a detailed account of Poincaré duality in noncommutative geometry. In this section, we simply give the axiom of Poincaré duality, as stated in [7], and derive its formulation in the finite case. The Poincaré duality axiom is

Poincaré duality
The intersection form K * (A) × K * (A) → Z is non degenerate.
Some insights on V are given by the knowledge of the vector bundles over V through its K-theory that can be formulated algebraically. K * (A) is a collective notation for all K-groups. Due to Bott periodicity, the sequence of these groups is periodic with period 2, for complex algebras. In the case of real algebras, the period is 8 [11]. However, if we do not take into account the torsion (i.e. the presence of K-groups of type Z/2Z), only matters the group K 0 (A). Since K 0 is additive, that is, we have to compute K 0 (M n (K)) [12]. By definition, K 0 (M n (K)) is the Grothendieck group of stably isomorphic classes of finitely generated projective modules over A. Recall that a module over M n (K) is a vector space E equipped with a representation of M n (K) which is said to be finitely generated if any of its elements can be written as a linear combination with coefficients in M n (K) of a finite number of elements of E. The additive structure is given by the direct sum of modules, and two such modules are said to be stably isomorphic if they are isomorphic after addition of a trivial module (i.e. a module that is simply a direct sum of several copies of the algebra).
The direct sum of modules yielding only an addition, we have to define the opposite of a module to obtain a group. This is achieved by means of the Grothendick procedure, which is used to construct the additive group of integers Z out of the positive integers N. Thus, we end up with an abelian group, which turns out to be the free group generated by the class of the "smallest" modules over M n (K). The latter is given by any non zero projection in M n (K) of minimal rank, since two projections of same rank are obviously equivalent and since the rank of the direct sum of two projections is the sum of their ranks. Now, we simply have to find such a projection in the real, complex and quaternionic cases. In the first two cases, the solution is obvious since any matrix whose only non vanishing entry on the diagonal is equal to 1 is a projection of minimal rank. Therefore such a projection has trace 1. In the quaternionic case, it remains true if we consider that the only non vanishing entry is the unity of H, so that it has trace 2. Finally, K 0 (A) turns out to be Z N and is the free group generated by (p i ) 1≤i≤N , where p i ∈ M n i (K) is a self-adjoint projection of minimal rank. As an additive group, it can be considered as a Z-module with basis (p i ) 1≤i≤N .
To formulate Poincaré duality for finite spectral triples, we have to compute the intersection form ∩ in this case. More precisely, since ∩ is a Z-bilinear form on K 0 (A) × K 0 (A), we have to compute its matrix given by ∩ ij = ∩(p i , p j ). To proceed, we apply the the general definition of the intersection form as the pairing of the Dirac operator index with the two projections p i and p j . Finite spectral triples are even spectral triples, so that we rewrite the Dirac operator as M is a map from the space of right-handed fermions (eigenspace of χ associated to the eigenvalue +1) to the space of left-handed fermions (eigenspace of χ associated to the eigenvalue -1), and can be considered as a mass matrix. Using the reality axiom, the Hilbert space is endowed with a A ⊗ A op -module structure that allows us to construct the linear map Both spaces being finite dimensional, the index is easily computed and we get, Since χ commutes with π(p i ) and J π(p j )J −1 , the operators 1/2π(p i )J π(p j )J −1 (1 + χ) and 1/2π(p i )J π(p j )J −1 (1 − χ) are projections. Therefore, the dimension of H R ij and H L ij are simply the trace of the corresponding projections. Finally, ∩ ij , defined as the pairing of p i ⊗ p j with the index of D, is given by Accordingly, we have obtained a simple formulation of Poincaré duality for the finite spectral triples that can be stated as follows.

Poincaré duality (finite case)
The matrix defined by ∩ ij = Tr [χ (π(p i )J π(p j )J −1 )] has non vanishing determinant, where p i ∈ M n i (K) is a self-adjoint projection of minimal rank.

First order condition
In ordinary differential geometry of a compact manifold endowed with a spin structure, the Dirac operator is a first order differential operator. This can be easily formulated using commutation relations. Indeed, let V be a compact manifold and E and F be smooth vector bundles over V . The space of sections Γ(E) and Γ(F ) of the vector bundles turn out to be modules over C ∞ (V ) and one can easily characterize first order operators. They are the differential operators for any couple of smooth functions f, g ∈ C ∞ (V ). This can also be extended to higher order differential operators: a differential operator D is of order n if only if, for any However, this notion cannot be extended in a direct manner in the noncommutative setting. In fact, we have to replace the modules by bimodules, and the functions in the double commutator by the left and right actions of the algebras. Since the latter commute, no matter whether the left or the right action appears first.

First order condition
We can rewrite the previous relation using bimodule notations. It becomes for any ψ ∈ H and x, y ∈ A. This is the usual definition of a first order operator on a bimodule [13].

S 0 -reality
The last four axioms define finite spectral triples. However, it turns out that the spectral triple of the standard model belongs to the smaller class of finite spectral triples that exclude Majorana particles. The operator J can be considered as a charge conjugation operator and it is natural to ask whether there are or not Majorana fermions. Spectral triples without Majorana fermions, such as the spectral triple of the standard model, are called S 0 -real spectral triples. They are defined by imposing a fifth axiom called.
There is a hermitian involution γ that commutes with χ, D and π(x) for any x ∈ A and anticommutes with J .
Since γ is a hermitian involution, one has γ * = γ and γ 2 = 1, so that one can decompose H into a direct sum of eigenspaces of γ. The eigenspace corresponding to eigenvalue +1 (resp. -1) is spanned by particles (resp. antiparticles). The anticommutation rule between γ and J teaches us that there is no particle that is its own antiparticle. In that case, H is a direct sum, where P and A, refer to particle and antiparticle and L and R to left and right. Within this decomposition, π and χ are block diagonal, . In a suitable basis, J and D are given by where C denotes the complex conjugation and M ∈ M n L ×n R (C) is a mass matrix. The reality and first order axioms are simply for any x, y ∈ A. Accordingly, particle and antiparticle spaces yield two bimodules, the latter being the opposite of the former in the sense that we exchange, up to complex conjugation, the left and right actions. Moreover, the Dirac operator, when restricted to the particle bimodule, is a first order operator, and it acts on antiparticle space as the complex conjugate. Therefore, the study of a S 0 -real spectral triple can be reduced to study of the particle bimodule.

General solution
The axioms of noncommutative geometry turn out to be very simple in the finite case. Moreover, the algebra A is a direct sum of matrix algebras and it becomes possible to give a general solution to these axioms.

The representation
We first assume that the algebra is made of complex matrices with a complex representation, i.e. a representation linear over the complex numbers, and then, we extend our results to the real case. Let A be a direct sum of complex matrix algebras, To fix notations, we write elements x, y of the algebra as N-uples: The only irreducible representation of M n (C) as a complex associative algebra is the fundamental one, so that π can be reduced as where, for m ∈ N, I m denotes the identity of M m (C). If m = 0, we assume that I m is equal to zero and does not appear in the previous decomposition.
The reality axiom provides us with two commuting representatioms of A, x → π(x) and y → J π(y)J −1 , that can be simultaneously decomposed into irreducible ones. We end up with a matrix of multiplicities, m ∈ M N (N), and we have To this reduction of π is associated the corresponding decomposition of H as a direct sum of mutually orthogonal subspaces, The action of π(x) and J π(y)J −1 on H ij are respectively x i ⊗I m ij ⊗I n j and I n i ⊗I m ij ⊗y j . Accordingly, H ij can be identified, as a bimodule, with m ij copies of M n i ×n j (C), where the left and right actions are simply left and right matrix multiplications.
In the real case, the situation is a bit more tricky. The algebras M n (R) and M n (H) have only one irreducible representation, the fundamental one, whereas M n (C) has two inequivalent irreducible representations, the fundamental one and its complex conjugate. However, all our formulae remain valid, provided the indices i and j label irreducible representations of A. In this case, x i denotes the image of x ∈ A by the irreducible representation corresponding to i, and H ij is isomorphic to C n i ⊗ C m ij ⊗ C n j , where n i is the dimension of the irreducible representation i.
To introduce the matrix of multiplicities m, we only use the bimodule structure of H. Thus, to any (A, B)-bimodule M, where M is finite dimensional and A and B are direct sums of matrix algebras, we associate a matrix of multiplicity m by reducing simultaneously the left and right actions. The row index of m labels the irreducible representations of A whereas its column index labels those of B. Furthermore, it is possible to perform some algebraic operations on the matrices of multiplicities: This is a very simple case of the theory of composition of correspondences, where A, B and C are general Von Neumann algebras [4,14]. It applies in particular for a S 0 -real spectral triple. The particle space is a bimodule associated to a matrix e, the antiparticle space is associated to e * and the total space corresponds to m = e + e * , so that the matrix of multiplicities is symmetric.
Conclusion A bimodule over a couple of semisimple involutive algebras is given, up to unitary equivalence, by a matrix of multiplicities whose entries are positive integers and whose row and column indices label irreducible representations of the corresponding algebras.

The charge conjugation
Given a (A, A)-bimodule H with matrix m, it is natural to ask whether it corresponds to a real structure (that is, we can pass from one representation to the other by means of a charge conjugation J as above) or not. Furthermore, if J exists, it is interesting to determine its most general form.
To answer these questions, let us first assume that it exists. The action of J on ψ ∈ H can always be written, in any basis, as J ψ = Kψ, where K is a matrix. J is an antiunitary involution if only if KK = KK * = 1. We must also have for any x ∈ A. Accordingly, Therefore, the dimensions of H ij and H ji are equal, or equivalently, the matrix of multiplicities m is symmetric: m ij = m ji . Let us denote by K ij : H ij → H ji the restriction of K. It must fulfill If i = j, the spaces H ij and H ji are distinct and it is always possible, by, a suitable choice of the orthonormal basis (ψ p ij ) 1≤p≤m ij , to have L ij ψ p ij = ψ p ji . When i = j, L ii is unitary, and we can choose an orthonormal basis such that L ii ψ p ii = e iφ p i ψ p ii , with φ p i ∈ R. However, since J is antilinear, so that the phases disappear if we make the replacement ψ p ij → e iφ p i /2 ψ p ij . Thus the action of J is simply in all cases. Conversely, if the matrix of multiplicities m is symmetric, it is possible to define J as above and such a antilinear operator fulfills all conditions imposed to charge conjugation.
In the S 0 -real case, we have already proved that m = e + e * . In this case, we have two bimodules, one with matrix e and the other with matrix e * . If we define J as the antilinear map that exchange the corresponding vectors of the two bimodules and γ the hermitian operator that takes value 1 on the first bimodule and −1 on the second one, they anticommutes and the direct sum of the two bimodules corresponds, at the bimodule level (i.e. if we do not take care of χ and D), to a S 0 -real structure.
Conclusion: a (A, A)-bimodule with matrix of multiplicities m is endowed with a real structure if only if m is symmetric. Moreover, if so, the charge conjugation is unique up to unitary equivalence, and it acts as above on tensor products. It corresponds to a S 0 -real structure if only if m can be written as m = e + e * , with e ∈ M N (N), or equivalently, if all diagonal entries of m are even.

The chirality
The chirality χ commutes with π(A) and J π(A)J −1 , and is hermitian, so that it can be written as a block diagonal matrix, Since χ = p π(x p )J π(y p )J −1 , with x p , y p ∈ A, χ ij is scalar matrix. Furthermore χ ij = ±1 as follows from χ 2 = 1. We also have [χ, J ] = 0 so that χ ij = χ ji . All the information contained in the matrices χ ij and m ij can be cast in a symmetric matrix µ ∈ M N (Z) defined by µ ij = χ ij m ij .
We recover χ ij and m ij as the sign and the modulus of µ ij . Accordingly, To proceed further, let us assume that we are dealing with the complex case. Thus all projections involved in Poincaré duality are of trace 1, and the matrix of the intersection form ∩ ij is given by According to Poincaré duality, the matrix µ must be non degenerate.
Conversely, if we start with a symmetric and non degenerate matrix µ ∈ M N (Z) (from now on we refer to it as the matrix of multiplicities instead of m), we recover the bimodule structure out of |µ ij | and the chirality χ is obtained as It fulfills all the requirements imposed by Poincaré duality and by the orientability axiom. For instance, let us check that there are some elements, x p and y p , of A such that χ = p π(x p )J π(y p )J −1 . To proceed, let us expand the matrix χ ij = sign(µ ij ) over rank one matrices as Thus we have χ = p π(x p )J π(y p )J −1 , with x p = (x p i 1 i ) 1≤i≤N and y p = (y p i 1 i ) 1≤i≤N . Accordingly, in the complex case, the bimodule structure and the chirality of a finite spectral triple are given, up to unitary equivalence, by a symmetric and non degenerate matrix µ with entries in Z, which is the matrix of the intersection form. Furthermore, it corresponds to a S 0real structure if only if the intersection form is an even quadratic form. In the real case, it is still given by a symmetric matrix with entries in Z whose indices label irreducibles representations of A. However, it is no longer the intersection form and the signs of its entries satisfy additional requirements that are summarized, together with its relation with the intersection form, in table 1. Table 1 In this table, the indices i and j label, for matrices with entries in a given field, the simple algebras that appear in the decomposition of A. In the complex case, the indices α and β distinguish a representation from its complex conjugate.

Conclusion:
The bimodule structure and the the chirality of a finite spectral triple are given, up to unitary equivalence, by a symmetric and non degenerate matrix of multiplicities. The moduli of the entries of the matrix of multiplicities determine the bimodule structure and their signs yield the chirality. To prove it, we decompose the representations associated with the left and right actions into irreducible ones and determine the corresponding matrix elements of D.

The Dirac operator
In particular, this applies to the Dirac operator D of a finite spectral triple that can be written as Moreover, since D anticommutes with χ, so do D L and D R . Accordingly, the previous decomposition is unique. Indeed, if we have, with obvious notations, R commutes with both π(A) and J π(A)J −1 . Thus, it commutes with χ, and since it also anticommutes with χ, it vanishes and the previous decomposition is unique. Furthermore, D commutes with J , so that, using the unicity, we have Consequently, the Dirac operator of a finite spectral triple is given by where ∆ is a hermitian operator on H, which commutes with J π(A)J −1 and anticommutes with χ. For a S 0 real spectral triple, it is sufficient to take into account the particle bimodule. The Dirac operator can be written as a direct sum, where D P is a first order operator of the particle bimodule. Using the unicity, we have The decomposition of D remains valid for the bimodule structure of the particle space, Let us end this section by three remarks that are useful when this construction is applied to model building in gauge theory. First of all, we can give an explicit expression of ∆ in terms of the Dirac operator. Indeed, where dg is the Haar measure on the group G of unitary elements of A. Due to the translational invariance of the Haar measure, G dg π(g)Dπ(g * ), π(x) = 0, for any x ∈ A. Futhermore, it follows from the reality axiom and the first order condition that G dg π(g) [D, π(g * )] , J π(x)J = 0, for any x ∈ A. Thus, it is a decomposition of the Dirac operator into left and right module homomorphisms that both anticommute with χ. Using the unicity, we have This expression of ∆ is very useful since it means that ∆ is a 1-form.
Secondly, The operator ∆ satisfies very simple commutation and anticommutation relations and we can give its explicit form in terms of complex matrices. Let be the decomposition of the Hilbert where i and j label irreducible representations of A, and P ij H → H ij be the orthogonal projection onto H ij . We define the matrix elements of ∆ by The latter are given by ∆ kl ij = δ jl M ik,j ⊗ I n j , where M ik,j ∈ M µ ij n i ×µ kj n k (C) is such that M ik,j = M * ki,j if µ ij µ kj < 0, and zero otherwise. This gives us a complete description of the operator ∆ parametrized by complex matrices, and we recover the Dirac operator as D = ∆ + J ∆J −1 .
Thirdly, notice that most of the time, the Dirac operator D only appears in noncommutative geometry via the commutation relation [D, π(x)], x ∈ A. Since D = ∆ + J ∆J −1 and [J ∆J −1 , π(x)] = 0 for any x ∈ A, one can replace D by ∆ in any commutation relation with an element of the algebra. This simplifies, for finite spectral triples, many computations involving the Dirac operator, such as, for instance, the computation of the metric on the space of pure states of A and its pertubation by inner fluctuations [7].
Finally, we use this decomposition to compute the 1-forms. We recall that the space of 1-forms Ω 1 D (A) is defined by In particular, is a 1-form. Moreover, if ω ∈ Ω 1 D (A) and x, y ∈ A, we have π(x)ωπ(y) ∈ Ω 1 D (A) and Ω 1 D (A) is a bimodule and thus contains the bimodule generated by ∆. Conversely, if we express any element ω ∈ Ω 1 D (A) using commutators with D and then, replace D by ∆, we see that Ω 1 D (A) is included in the bimodule generated by ∆. Accordingly, the space of 1-forms Ω 1 D (A) is the bimodule generated by ∆, and this allows us, using the expression of ∆ in terms of complex matrices, to determine Ω 1 D (A) in section 5. Conclusion: A finite spectral triple is given, up to unitary equivalence by a symmetric and non degenerate matrix of multiplicities and a hermitian operator ∆ commuting with the right action and anticommuting with the chirality.

Classification and diagrams
In the following sections, we develop a diagrammatic approach to finite spectral triples. We first associate to any finite spectral triple a diagram and then, we classify and construct all such triples within this approach.

Diagram associated with a spectral triple
Let (A, H, D) be a finite spectral triple whose matrix of multiplicities is µ. To each couple (i, j) of irreducible representations of A such that µ ij = 0, we associate a vertex of type ⊖ if µ ij < 0 and a vertex of type ⊕ if µ ij > 0. We relate the vertices (i, j) and (k, l) if only if the corresponding matrix element of the Dirac operator, does not vanish. According to the axioms, the diagram has the following properties.
• (i, j) is related to (k, l) if only if (k, l) is related to (i,j) as follows from D = D * . To simplify, we draw only one edge between two vertices.
• Since D and J commute, the diagram is symmetric with respect to its first diagonal.
• We can only relate vertices of different types, as follows from the anticommutation relation of χ and D.
• As a consequence of the first order condition, all edges are either vertical or horizontal. Vertical edges correspond to matrix elements of ∆ and the horizontal ones to those of J ∆J −1 .
For instance, let us build the diagram pertaining to the standard model. The algebra A is Since it is S 0 -real, we use the notations introduced in section 2.5. The particle Hilbert spaces H P L and H P R are spanned, for N f = 3 families of fermions, by where we have omitted the color index for quarks. The corresponding antiparticles form a basis of the antiparticle spaces H A L and H A R . Within these bases, the representation is given by where m p stands for the mass of particle p and V CKM is the Cabibbo-Kobayashi-Maskawa mixing matrix. Accordingly, the matrix of multiplicities is, in the basis (C, H, C, M 3 (C)), and the corresponding diagram is given by figure 1. For particles, vertices and edges are above the first diagonal. The three vertices ⊕ correspond to the singlets of isospin u R , d R and e R and the two vertices ⊖ are associated with the leptonic Figure 1: Diagram of the standard model and quarkonic isodoublets. The edges between them correpond to the matrix elements M u , M d and M e . Since it is S 0 -real, there is a complete symmetry between particles and antiparticles so that the same holds for the antiparticles. In fact, for a general S 0 -real spectral triple, the matrix of multiplicities can always be written as The Dirac operator is a first order operator of the particle bimodule and the complex conjugate on the antiparticle bimodule. Thus, the corresponding diagram is constructed out of the vertices of the matrix ǫ and then symmetrized with respect to the first diagonal.
The diagram of the standard model is not connected and has four connected subdiagrams that are intimately tied up with the "noncommutative coupling constants" x, y,x andỹ, introduced in [15]. In general, the latter is a positive definite hermitian matrix Z that commutes with π(A), J π(A)J −1 and D, so that it yields a scalar product on the differential algebra. The first two commutation relations allow us to write Z as a block diagonal matrix, Accordingly, to each vertex (i, j) is associated a positive definite hermitian matrix Z ij . The last commutation relation, expressed using matrix elements as relates the matrix elements Z ij associated with vertices belonging to the same connected component.
The diagrams provides us with a natural classification of finite spectral triples. As we shall see in the sections devoted to applications, the major feature of the underlying Yang-Mills-Higgs model can be seen on the diagram. However, a commutative spectral triple, with the algebra C N and a noncommutative one, whose algebra is a direct sum of N simple matrix algebras, can yield the same diagram. From the point of view of noncommutative geometry, this is not surprising since two such algebras are Morita equivalent and thus completely indistinguishable in Hoschild homology and K-theory. Besides, using the general construction, based on a connection ∇ on a finite projective module E, detailed in [7], we can pass from a commutative spectral triple to a noncommutative one which are both associated with the same diagram.

Spectral triple associated to a diagram
Suppose we are given an algebra A and a symmetric matrix of multiplicities µ whose indices label irreducible representations of A. We require that it fulfills the constraints listed in table 1 and that its intersection form ∩ ij be non degenerate. We represent it on the plane as the vertices of a diagram as we did in the previous section. Then, we relate the vertices of different types by vertical and horizontal lines such that the resulting diagram be symmetric with respect to its first diagonal.
To each vertical link between (i, k) and (j, k), with i > j, we associate a matrix M ij,k ∈ M µ ik n i ×µ jk n j (C). If j > i, we set M ji,k = M * ij,k and M ij,k = 0 in all other cases. The matrix elements of the operator ∆ are obtained as In particular, the diagonal elements ∆ ij ij always vanish since we cannot a relate a vertice to itself. We symmetrize this expression with respect to J to get the complete Dirac operator To obtain an explicit form of D in terms of complex matrices, we introduce a basis (E p ij ) 1≤p≤n i n j , for example the canonical one, and we write M ij,k as Thus, the matrix elements of the Dirac operator are The representation, the Hilbert space and the operators J and χ are constructed out of the matrix of multiplicities by means of the formulae we gave throughout the sections 3.1-3.3.

Applications to particle models
Finite spectral triples are very simple examples of noncommutative geometries corresponding to finite spaces. The latter can be considered as the internal spaces involved in Yang-Mills-Higgs theories. More precisely, if we tensorize a finite spectral triple with the one associated to an ordinary manifold of dimension n, we obtain a noncommutative geometry of dimension n that gives rise to a Yang-Mills theory with spontaneous symmetry breaking. This section is devoted to this kind of applications. We first recall some general aspects about gauge theory in noncommutative geometry and then apply our previous classification to investigate some features of the model we obtain.

General aspects
It follows from general principles of noncommutative geometry [7], that the symmetries of a spectral triple (A, H, D) are described by * -automorphisms of the algebra A. Among them, there are, in the noncommutative case, inner automorphisms. The latter are defined, as x ∈ A → gxg −1 ∈ A, for any element g of the group G of unitary elements of A. If the algebra is finite dimensional, the inner automorphisms are the only automorphisms of A up to a discrete subgroup. Thus, they are the only possible continuous symmetries of a finite noncommutative space. To describe a symmetry of the corresponding space, the group G is represented on the Hilbert space H via the map g → π(g)J π(g)J −1 . The inner automorphisms are in complete analogy with gauge transformations In the framework of noncommutative geometry, H is the fermionic Hilbert space and D the Dirac operator. Accordingly, the free fermionic action is defined as for any Ψ ∈ H. Under a gauge transformation, Ψ −→ π(g)J π(g)J −1 Ψ, the fermionic action becomes where we have used the axioms to rewrite To ensure gauge invariance, we introduce, as usual in field theory, a gauge field A. The latter are hermitian 1-forms and can be written as with x p , y p ∈ A. The transformation law of A is A → π(g)Aπ(g * ) + π(g) [D, π(g * )] , so that the interacting fermionic action is gauge invariant. Starting with the spectral action principle, "the bosonic action only depends on the spectrum of D + A + J AJ −1 ", it is possible to write a bosonic action for the field A.
With a suitable spectral triple, it yields, as first terms of heat kernel expansion, the bosonic sector of the standard model coupled to gravity [9].
To construct a Yang-Mills-Higgs theory, we consider a noncommutative geometry (A t , H t , D t ) obtained as a product of the ordinary geometry of space time by a finite noncommutative one, described by a finite spectral triple (A, H, D), that corresponds to the internal degrees of freedom of the fields. (A t , H t , D t ) is a tensor product of spectral triples, where ω µ is the spin connection introduced to take into account the coupling of fermionic fields with gravity. Therefore, the gauge fields A t are given by where A µ is a vector field with values in the Lie algebra G of the group of unitary elements of A, and H is a scalar field that is also a hermitian 1-form with respect to the finite geometry. The corresponding fluctuations of the metric [7] are Using the decomposition D = ∆ + J ∆J −1 , the scalar part can be written as with Φ = ∆ + H. Moreover, since ∆ is a 1-form, Φ is a gauge field and it transforms as Φ −→ π(g)Φπ(g * ).
The interaction of fermions with scalars can be written using Φ as Thus, Φ is a matrix valued scalar field that contains all Higgs fields, and the previous interaction term corresponds to Yukawa couplings.
Finally, following the spectral action principle, the Higgs potential V (Φ) only depends on the spectrum of Φ + J ΦJ −1 and thus can be expanded as In dimension four, we restrict our study to polynomial potential of degree ≤ 4, so that where a 0 , a 2 and a 4 are real numbers related to other parameters of the theory using the spectral action principle applied to the full Dirac operator.

The gauge group
The gauge transformations correspond to the inner automorphisms of the finite dimensional algebra A. Accordingly, the gauge group is the group G of unitary elements of A. Since A is a direct sum of matrix algebras over the fields R, C and H, the gauge group is a direct product of the simple Lie groups SO(n), U(n) and SP (n). As a consequence, noncommutative geometric models can have all classical semi-simple compact Lie group as a gauge group, except those corresponding to the five exceptional Lie algebras [16].
However, the gauge group only corresponds to G up to some U(1) factors. Indeed, the elements of G appear in gauge transformations as π(g)J π(g)J −1 so that we have to divide G by the kernel of the representation g → π(g)J π(g)J −1 to obtain the gauge group. The latter only contains U(1) factors, that corespond to the U(1) whose representations only appear on diagonal vertices in the diagram associated to (A, H, D). These U(1) gauge fields also disappear from the action because the action only depends on π(A µ ) + J π(A µ )J −1 and they are the solutions of the equation π(A µ ) + J π(A µ )J −1 = 0.
For example, let us consider a model with fermions in the adjoint representation. It is built with A = M n (C) and H = M n (C). The representation of A on H is the left multiplication of matrices and J is given by the adjoint. Obviously, it corresponds to a diagram with a single vertex so that its gauge group is SU(n) instead of U(n) = SU(n)×U(1). Moreover, from vertex uniqueness, we see immediately that it can only afford fermions with same chirality and there is no possible link, thus no Dirac operator and no Higgs field. Corresponding to the simplest diagram, this model can be considered as the minimal one. It can be extended to semi-simple algebras, with bimodule structure given by left and right multiplication. It yields a diagonal matrix of multiplicities and a diagonal diagram. Accordingly, all U(1) disappear but there is still no possible links, thus no Higgs fields and no spontaneous symmetry breaking. For the standard model (see Figure 1), the diagram has no diagonal elements. Therefore, the unwanted U(1) in the group of unitary elements of H ⊕ C ⊕ M 3 (C) does not disappear and has to be eliminated by hand.
In more sophisticated models, we often have to remove by hand some U(1) factors for physical reasons such as anomaly cancellation (of course, it may also be necessary to remove non-abelian Lie subgroups but this will change too much the noncommutative geometric setting of the model). This is achieved by means of a unimodularity condition, that can be defined as a set of linear conditions imposed at the Lie algebra level in order to eliminate the unwanted U(1) factors. For instance, if we want our model to be gauge anomaly free, we impose that the generators of the Lie algebra of the gauge group, satisfy the cubic relation Tr T a T b , T c = 0.
Since the representation we deal with are rather simple (tensor products of fundamental ones or their complex conjugates), the gauge anomaly cancellation can be expressed as constraints on the matrix of multiplicities of the model. For instance, let us consider a S 0 -real spectral triple with the complex algebra and a complex representation. Taking into account the occurence of left and right-handed fermions, the condition of anomaly is simply where G is the Lie algebra of the gauge group and Tr p denotes the trace restricted to particle space. To express it in terms of the matrix of multiplicities, we write any x i ∈ u(n i ) (Lie algebra of U(n i )) as x i = λ i + u i , with λ i ∈ su(n i ) and u i ∈ u(1). Since we only want to remove some U(1) factors, we require that this condition be satisfied at least for all non abelian factors. It yields three different type of constraints on the matrix of multiplicities ǫ.
The first equations are rigid constraints that must be satisfied by the matrix of multiplicities. Ths second ones are linear equations for the u i 's so that they may be taken as unimodularity conditions. The last one is a cubic relation on the "hypercharges", as in the standard model. It is a well known fact that for the standard model, anomaly cancellation is equivalent to the unimodularity condition [17]. In other words, if the second relations are satisfied, the third one also is. Obviously, this is not a general feature of the models. Moreover, for the standard model, the first relation is empty since the color sector is vectorial. We derived these relations for complex representations of complex algebras, but their general structure remains unchanged in the real case, since unitary elements of real and quaternionic matrices are traceless as well as their cubes. We still end up with three types of constraints: some rigid ones on the matrix of multiplicities, some linear equations for the U(1) factors and a cubic relation for the "hypercharges".
There is an other reason to remove an unwanted U(1). Since the Hilbert space must be equipped with a representation of an associative algebra, it is not possible, in general, to build a singlet under a given gauge group. To proceed, we introduce an additional summand C in the algebra, so that this singlet sits in a representation of the associative algebra C. Then, we remove the corresponding U(1) factor from the gauge group and we end up with a singlet under gauge transformations.

The fermionic representation
Within the diagrammatic approach, to each couple (i, j) of irreducible representations is associated a vertex (i, j) of the diagram. This vertex corresponds to the Hilbert space H ij appearing in the decomposition of H. H ij is endowed with a bimodule structure given by i (left action) and j (right action). The restriction of π(g)J π(g)J −1 to H ij is g i ⊗ I |µ ij | ⊗ g j , so that vectors of H ij transform in the tensor product i ⊗ j of the corresponding representations i and j of the simple factors of gauge group, with multiplicity |µ ij |. If the two factors are distinct, the tensor product is irreducible as representation of G and, for each simple factor it is just the fundamental one or its complex conjugate. Otherwise, the tensor product is not irreductible. For SO(n) and SP (n), the tensor product of two fundamental representations yields the adjoint one, the symmetric and traceless one and a singlet. For SU(n) the situation is different, since we can tensorize the fundamental one and its complex conjugate, and we decompose the resulting product using Young boxes. In the analysis, we do not care about the unimodularity conditions. However, the latter are simply linear conditions imposed on the U(1), and do not change the general strucure of the fermionic representation.
Moreover, to each vertex is associated a sign corresponding to the eigenvalue of χ on H ij . Accordingly, fermions of H ij are chiral ones, right-handed if µ ij > 0 and left-handed if µ ij < 0.  Many general properties of the fermionic sector, such as chirality, charge conjugation and transformation laws can be read on the diagram. As an example, let us try to build the SU(5) grand unified model. To proceed, we need two representations of SU(5): the 5 for right-handed fermions and the 10 for left-handed ones. The 5 can be obtained with the algebra M 5 (C) ⊕ C and the corresponding bimodule structure is given by the couples (5, 1) or (5, 1). At the gauge group level, we remove the unwanted U(1) with a unimodularity condition, as well as the one arising from M 5 (C). If we tensorize the 5 by itself we get the 10 and the 15. Thus, if we remove the fermions sitting in the 15, it works perfectly well from the point of view of representation theory. In the basis composed by the representations M 5 (C), M 5 (C) and C, the matrix of multiplicities reads, with N f generations, The corresponding diagram is given in figure 2.
To end this section, let us note that it is impossible to build the SO(10) model out of a spectral triple. Indeed, it requires the 16 (spinorial representation of SO(10)) that cannot be obtained as a subrepresentation of a tensor product of the fundamental one or its complex conjugate.

The scalar fields and their representations
To determine the Higgs fields and their transformation laws, we first have to compute explicitely the space of 1-forms. We recall that ∆ is a 1-form and that the space of 1-forms Ω 1 D (A) is the bimodule generated by ∆. Thus, if (E p ij ) 1≤p≤n i n j is a basis of M n i ×n j (C), the 1-forms span the space Ω 1 D (A). However, they are not, in general, linearly independent. To find a basis of Ω 1 D (A), we must have a closer look at the matrix structure of ∆. To simplify our discussion, let us assume that we are dealing with a complex spectral triple. In this case, it is useful to choose (E p ij ) 1≤p≤n i n j as the canonical basis of M n i ×n j (C). We denote it by E ab ij , and the multiplication rule is simply As in section 4.2, we rewrite the matrix elements of ∆ as To construct a basis, let us consider the matrices (M ab ij,k ) 1≤k≤N as a column vector M ab ij . The vectors M ab ij span, when a and b vary, a vector space V ij whose dimension is denoted by r ij . Out of the vectors (M ab ij ), we choose a basis of V ij that we denote by (M p ij ) 1≤p≤r ij . The components M p ij,k of M p ij yield a basis θ ab,p ij of Ω 1 D (A) defined by Accordingly, Ω D 1 (A) is a vector space of dimension ij n i r ij n j , and it can be parameterized by complex matrices as In the real case, it works in much the same way if i and j label irreducible representations of A. However, in the quaternionic case we must tensorize the elementary matrices with Pauli matrices. We also have some additional constraints between the matrices θ ij such as θ ij = θ ij , where i denotes the representation complex conjugate of i and θ is either the complex or the SU(2) conjugation. This applies in particular for the Higgs field Φ that can be parameterized by matrices φ p ij as φ = ijkp P * ik φ p ij ⊗ M p ij,k ⊗ I n k P jk .
Since Φ is hermitian, we have φ p ij * = φ p ji . Under a gauge transformation, φ −→ π(g)Φπ(g −1 ), the matrix valued scalar fields φ p ij transform as where, as usual, g i denotes the image of g through the irreducible representation labelled by i. Accordingly, the fields φ p ij transforms as the tensor product i ⊗ j of the two irreducible representations i and j.
On the diagram, the fields φ p ij correspond to vertical links between the rows i and j. Since such links cannot occur within the same row, the Higgs fields do not transform as the tensor product i ⊗ i, but all other tensor products are allowed. In particular, we never end up with Higgs fields in the adjoint representation. Therefore, it is not possible to build the conventional SU(5) grand unified model in our framework. Although we proved that it can work at the level of the fermionic representation, to break SU(5) down to SU(2) × U(1) × SU(3), we need some Higgs fields sitting in the adjoint representation of SU (5).
The Yukawa coupling between Φ and Ψ, can be expressed in terms of the fields φ p ij if we expand Ψ over the tensor products we introduced in section 3.2. As a result, it appears the Yukawa couplings between the chiral fermions of the vertices (i, k) and (j, l) and the scalar fields φ ij or φ kl correspond to the vertical or horizontal edges that can occur between the vertices (i, k) and (j, l).
According to the spectral action principle, the Higgs potential is a linear combination of terms like Tr Φ + J ΦJ −1 n .
Using the explicit form of Φ in terms of the Higgs fields φ q ij , the previous function can be expanded over monomials like Tr φ p 1 i 1 i 2 ...φ pr iri 1 Tr φ q 1 j 1 j 2 ...φ qs jsj 1 , with r + s = n. The coefficient of this monomial in the Higgs potential depends on the matrices M p ij,k . It corresponds canonically to a loop of length n contained in the diagram, made of r horizontal and s vertical edges. When we are dealing with loops, we must be aware that we have actually two links between the vertices (i, k) and (j, l), one for M ij,k and the other for M ji,k , even if, for simplicity, we represent only one. For instance, the sequence of vertices, (i, k) → (j, k) → (i, k), must be considered as a loop. We refer to this kind of loops, obtained by going forwards and backwards along a tree, as trivial loops. The previous example corresponds, at the level of the Higgs potential, to the monomial which is a mass term. In general, mass terms only corresponds to trivial loops. Non trivial ones, such as the sequence (i, k) → (j, k) → (j, l) → (i, l) → (i, k), with i = j and k = l, can only occur for n ≥ 4, that is, for interacting terms.
Starting with the notion of finite spectral triple derived from general axioms, we give all possible noncommutative geometries on finite spaces. The latter are given, up to unitary equivalence, by a symmetric and non degenerate matrix of multiplicities µ with entries in Z and an operator ∆. The underlying bimodule strucrure is encoded in the modules of the entries of the matrix of multiplicities whereas their signs correspond to the chirality. The matrix is symmetric if only if it corresponds to a real structure, and it follows from Poncaré duality that it is non degenerate. Finite spectral triples are classified and constructed using diagrams whose vertices correspond to their matrix of multiplicities and whose edges represent the structure of the Dirac operator. Finite spectral triples are also tools to build some Yang-Mills-Higgs models, and a general feature of the resulting models can be summarized as follows: • The gauge group is, up to some U(1) factors, a direct product of the simple Lie groups SO(n), SU(n) and SP (n).
• Fermions transform as tensor products of two fundamental representations (or their complex conjugate) of the simple factors. Multiplets of chiral fermions correspond to the vertices of a diagram labelled by couples of fundamental representation or their complex conjugate.
• The edges of the previous diagram correspond naturally to Yukawa couplings. The resulting Higgs fields also transform as tensor products of two fundamental representations (or their complex conjugate)of the simple factors of the gauge group, except those leading to the adjoint one.
• The Higgs potential, computed with the spectral action principle, can be expanded over monomials that correspond to loops of the diagram.
This study confirms, in a more general setting, some earlier results pertaining to the standard model [18], its U(1) extensions [19] and grand unified theories [20]. Furthermore, such a classification has also been worked out independently by M. Paschke and A. Sitarz in [21], putting emphasis on Hopf algebra symmetries.