In order to build, in non commutative geometry, a generalized gauge theory model, or even something very elementary like the notion of generalized covariant differential, one needs the following three ingredients. 1) An associative algebra (take for instance or . 2) A module for (choose any one you like). 3) A differential -graded algebra that will replace the usual algebra of differential forms (De Rham complex). The choice of the last ingredient is not at all unique. For instance, one can take for :
1. The algebra of universal differential form on (one can always do so!). The differential algebra of universal forms on is where is the kernel of the multiplication map, therefore, as a vector space, . Set , since p=3. We see that is of rank n-1 as a -module and of complex dimension . More generally, , so that has rank as a module and is of complex dimension .
2. The algebra of -valued antisymmetric forms on the Lie algebra of derivations of , which are linear w.r.t. the center of ( is usually not an module). This is the choice advocated by .
Understanding the structure of the Lie algebra and of its own representation theory is an interesting subject which we plan to return to in a separate work. We just recall here a few basic facts. First of all derivations of are all inner (they are given by commutators). The trace is therefore irrelevant and we can identify the Lie algebra of derivations of with . By imposing also a reality condition (hermiticity) one can obtain Lie(SU(3)). Suppose that one defines the algebra in terms of matrices as the linear span of elementary matrices and , it is easy to see that commutators with and define derivations that are not inner since these elements do not belong to , but they are valued in the module of matrices. In a version of non commutative differential calculus using , such derivations can be related to the notion of Higgs doublets. In the case of , which contains a Grassmann envelope (see first section), one has also to take into account the fact that the algebra is not semi-simple since it contains, in particular . Remember that derivations of Grassmann algebras are outer, and that, in particular, the vector space of graded derivations of a Grassmann algebra with two generators can be identified with the Lie superalgebra whose representation theory () is known to contain the representations that are needed to build the Standard Model of electroweak interactions (although the model is by no means obtained by gauging this superalgebra! See  and ).
3. When is the tensor product of by a finite dimensional algebra, one can also take as the tensor product of the usual De Rham complex, for M, times the algebra of universal forms for the finite geometry. In the case of this was the choice made in  (see also ). Here, keeping in mind applications to particle physics, one could take for the tensor product , where the first factor refers to the usual De Rham complex of differential forms over ``space-time''.
4. The algebra associated with a K-cycle on , i.e. the choice of a Hilbert space and a generalized Dirac operator D. This is the choice (``spectral triple'') advocated by  (and references therein).
In the present case, all of the above choices are possible, and also others, taking into account the existence of twisted derivations, etcSince we do not plan here to build any particular physical model, we stop here our discussion concerning the choice of the differential algebra .
The following observation was made, in 1990, by  and : They show that the n-power of a quantum SU(2) matrix with deforming parameter q is a quantum matrix with deforming parameter . This fact was then recovered and generalized in ,  . We describe this as follows. Let be a quantum matrix i.e., with q an arbitrary complex number, we assume that symbols a,b,c,d obey the six relations a b = q b a, b c = c b, a c = q c a, b d = q d b, c d = q d c and , a central element. We define then by
For instance , etc . One then shows that the six relations , etc are satisfied. Actually, one proves that , with , where et are operators obeying the relations , and . The result concerning powers of quantum matrices follows.
This result implies immediately that the ``algebra of functions on '' is a subalgebra of the ``algebra of functions on '' as soon as s divides r and that, in particular, the algebra of functions on the classical group is a subalgebra of as soon as q is a root of unity. This embedding, obtained by using properties of powers of quantum matrices is an embedding of algebra but not of coalgebras; this can be seen as follows (we compare and ): the coproduct on the algebra spanned by the coordinate functions generating reads, when applied to the generator a, , etcSince is an algebra homomorphism i.e. , whereas the Hopf algebra has another coproduct, namely equal to . Therefore and are usually different.
Warning: We already mentioned the fact that (more precisely ) can be considered as a Hopf subalgebra of provided we define it by using divided powers of the Chevalley generators. We do not know any relation between this kind of embedding, which can be generalized to other q-analogues of Lie simple groups  and ) and the algebra embedding mentioned above (which seems to be only valid for ).
Embedding of in (with q a root of unity) can be visualized as a projection from the quantum group to the classical one, with a finite quantum group as ``fiber''. This finite quantum group should therefore itself be thought of as a ``group'' included in . Despite of the free use of a terminology borrowed from commutative geometry, note that in the present situation, spaces have no points (or very few...)! Morally, one would like to replace the enveloping algebra of the Lorentz group by the quantum enveloping algebra of , when . At the intuitive level (and although theses spaces have very few points) one can see the classical Lorentz group as a quotient of the quantum , with q a primitive root of unity, by a ``discrete quantum group'' described by . This was the idea advocated in a comment of . We refrain to insist on the obvious similarities between some aspects of representation theory of and the Standard Model of elementary particles. We also refrain to insist on the obvious differences Notice that the finite quantum group , of dimension 27 (or , of dimension ) is an analogue of the discrete group that describes the relation between the Lorentz group and the spin group (the latter being the universal cover of the former): , relation which is, classically, at the origin of the difference between particles of integer and half-integer spin. Here, we have something analogous for . Whether or not one can build a realistic physical model along these lines, with non trivial prediction power, remains to be seen. We hope that the present contribution may help the interested readers to develop new ideas in this direction.
Many results to be found here came from discussions with Oleg Ogievetsky. I want to thank him for many enlightening comments and, in particular, for his patience in explaining me the basics concerning representation theory of non semi-simple associative algebras.
This work was supported, in part, by a grant from the Instituto Balseiro (Centro Atomico de Bariloche). I want to thank everybody there for their warm hospitality and for providing a peaceful atmosphere which made possible the writing of these notes.