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Other properties of tex2html_wrap_inline1123

Differential algebras.

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 tex2html_wrap_inline2181 (take for instance tex2html_wrap_inline2183 or tex2html_wrap_inline2185 . 2) A module for tex2html_wrap_inline2181 (choose any one you like). 3) A differential tex2html_wrap_inline2079 -graded algebra tex2html_wrap_inline2191 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 tex2html_wrap_inline2193 :

1. The algebra tex2html_wrap_inline2195 of universal differential form on tex2html_wrap_inline2181 (one can always do so!). The differential algebra of universal forms on tex2html_wrap_inline1123 is tex2html_wrap_inline2201 where tex2html_wrap_inline2203 is the kernel of the multiplication map, therefore, as a vector space, tex2html_wrap_inline2205 . Set tex2html_wrap_inline2207 , since p=3. We see that tex2html_wrap_inline2211 is of rank n-1 as a tex2html_wrap_inline1123 -module and of complex dimension tex2html_wrap_inline2217 . More generally, tex2html_wrap_inline2219 , so that tex2html_wrap_inline2221 has rank tex2html_wrap_inline2223 as a tex2html_wrap_inline1123 module and is of complex dimension tex2html_wrap_inline2227 .

2. The algebra tex2html_wrap_inline2229 of tex2html_wrap_inline2181 -valued antisymmetric forms on the Lie algebra of derivations of tex2html_wrap_inline2181 , which are linear w.r.t. the center of tex2html_wrap_inline2181 ( tex2html_wrap_inline2237 is usually not an tex2html_wrap_inline2181 module). This is the choice advocated by [15].

Understanding the structure of the Lie algebra tex2html_wrap_inline2241 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 tex2html_wrap_inline2243 are all inner (they are given by commutators). The trace is therefore irrelevant and we can identify the Lie algebra of derivations of tex2html_wrap_inline1407 with tex2html_wrap_inline2247 . By imposing also a reality condition (hermiticity) one can obtain Lie(SU(3)). Suppose that one defines the algebra tex2html_wrap_inline2251 in terms of tex2html_wrap_inline1171 matrices as the linear span of elementary matrices tex2html_wrap_inline2255 and tex2html_wrap_inline2257 , it is easy to see that commutators with tex2html_wrap_inline2259 and tex2html_wrap_inline2261 define derivations that are not inner since these elements do not belong to tex2html_wrap_inline2251 , but they are valued in the module of tex2html_wrap_inline1171 matrices. In a version of non commutative differential calculus using tex2html_wrap_inline2267 , such derivations can be related to the notion of Higgs doublets. In the case of tex2html_wrap_inline1123 , 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 tex2html_wrap_inline2271 . 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 tex2html_wrap_inline2273 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 tex2html_wrap_inline2181 is the tensor product of tex2html_wrap_inline2277 by a finite dimensional algebra, one can also take tex2html_wrap_inline2193 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 tex2html_wrap_inline2283 this was the choice made in [7] (see also [9]). Here, keeping in mind applications to particle physics, one could take for tex2html_wrap_inline2193 the tensor product tex2html_wrap_inline2287 , where the first factor refers to the usual De Rham complex of differential forms over ``space-time''.

4. The algebra tex2html_wrap_inline2289 associated with a K-cycle on tex2html_wrap_inline2181 , 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 tex2html_wrap_inline2191 .

Powers of SL(2) quantum matrices.

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 tex2html_wrap_inline2309 . This fact was then recovered and generalized in [22], [23] . We describe this as follows. Let tex2html_wrap_inline2311 be a quantum tex2html_wrap_inline1731 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 tex2html_wrap_inline2329 , a central element. We define then tex2html_wrap_inline2331 by


For instance tex2html_wrap_inline2335 , etc . One then shows that the six relations tex2html_wrap_inline2337 , etc are satisfied. Actually, one proves that tex2html_wrap_inline2339 , with tex2html_wrap_inline2341 , where tex2html_wrap_inline2343 et tex2html_wrap_inline2345 are operators obeying the relations tex2html_wrap_inline2347 , tex2html_wrap_inline2349 and tex2html_wrap_inline2351 . The result concerning powers of quantum matrices follows.

This result implies immediately that the ``algebra of functions on tex2html_wrap_inline2353 '' is a subalgebra of the ``algebra of functions on tex2html_wrap_inline2355 '' as soon as s divides r and that, in particular, the algebra of functions on the classical group tex2html_wrap_inline1731 is a subalgebra of tex2html_wrap_inline2363 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 tex2html_wrap_inline2363 and tex2html_wrap_inline2369 ): the coproduct on the algebra spanned by the coordinate functions tex2html_wrap_inline2371 generating tex2html_wrap_inline2373 reads, when applied to the generator a, tex2html_wrap_inline2377 , etcSince tex2html_wrap_inline1557 is an algebra homomorphism tex2html_wrap_inline2381 i.e. tex2html_wrap_inline2383 , whereas the Hopf algebra tex2html_wrap_inline2385 has another coproduct, namely tex2html_wrap_inline2387 equal to tex2html_wrap_inline2389 . Therefore tex2html_wrap_inline1557 and tex2html_wrap_inline2393 are usually different.

Warning: We already mentioned the fact that tex2html_wrap_inline2395 (more precisely tex2html_wrap_inline2033 ) can be considered as a Hopf subalgebra of tex2html_wrap_inline1157 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 tex2html_wrap_inline1143 ).

General remarks.

Embedding of tex2html_wrap_inline2405 in tex2html_wrap_inline2363 (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 tex2html_wrap_inline1615 . 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 tex2html_wrap_inline2413 by the quantum enveloping algebra of tex2html_wrap_inline1615 , when tex2html_wrap_inline1187 . 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 tex2html_wrap_inline1615 , with q a primitive root of unity, by a ``discrete quantum group'' described by tex2html_wrap_inline1123 . 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 tex2html_wrap_inline2425 and the Standard Model of elementary particles. We also refrain to insist on the obvious differences tex2html_wrap_inline2427 Notice that the finite quantum group tex2html_wrap_inline1123 , of dimension 27 (or tex2html_wrap_inline2433 , of dimension tex2html_wrap_inline1907 ) is an analogue of the discrete group tex2html_wrap_inline1175 that describes the relation between the Lorentz group and the spin group (the latter being the universal cover of the former): tex2html_wrap_inline2439 , relation which is, classically, at the origin of the difference between particles of integer and half-integer spin. Here, we have something analogous for tex2html_wrap_inline1123 . 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.

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Next: References Up: No Title Previous: Others avatars for and

Robert Coquereaux
Tue Nov 5 15:18:21 MET 1996