** Next:** Introduction

*CC 439 - 8400 -San Carlos de Bariloche - Rio Negro - Argentina*

*F-13288 Marseille Cedex 9 - France *

We describe a few properties of the non semi-simple associative algebra , where is the Grassmann algebra with two
generators. We show that is not only a finite
dimensional algebra but also a (non co-commutative) Hopf algebra, hence a finite
dimensional quantum group. By selecting a system of explicit
generators, we show how it is related with the quantum
enveloping algebra of when the parameter *q* is a cubic root of
unity. We describe its indecomposable projective representations as well
as the irreducible ones. We also comment about the relation between this object and
the theory of modular representations of the group
, i.e. the binary tetrahedral group. Finally, we briefly discuss
its relation with the Lorentz group and, as already suggested by
A.Connes, make a few comments about the possible use of this algebra in a
modification of the Standard Model of particle physics (the unitary group
of the semi-simple algebra associated with
is ).

**anonymous ftp or gopher**: cpt.univ-mrs.fr

**Keywords**: quantum groups, Hopf algebras, standard model,
particle physics, non-commutative geometry.

**hepth**: 9610114

CPT-96/P.3388

- Introduction
- The algebra
- A system of generators for
- The coproduct on
- The representation theory of
- Others avatars for and related algebras
- Other properties of
- References
- About this document ...

Tue Nov 5 15:18:21 MET 1996