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 [15].
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 ([24]) 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 [26] and [8]).
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
[7] (see also [9]). 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 [5] (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 [31]
and [11]: 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 [22],
[23]
. 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 [20] and
[21]) 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
[4].
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.
Acknowledgments
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.