Exceptional quantum geometry and particle physics

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space $\mathbb C\oplus\mathbb C^3$ is associated to the quark-lepton symmetry, (one complex for the lepton and 3 for the corresponding quark). More generally it is is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras, (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.Comment: 37 pages ; some minor typo corrections. To appear in Nucl. Pays. B (2016), http://dx.doi.org/10.1016/j.nuclphysb.2016.04.01

Some aspects of noncommutative differential geometry

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework.Comment: 27 pages, AMS-LaTeX, LPTHE-ORSAY 95/7

The quantum group of a preregular multilinear form

We describe the universal quantum group preserving a preregular multilinear form, by means of an explicit finite presentation of the corresponding Hopf algebra.Comment: 17 pages - Ref. 11 correcte