next up previous

On the finite dimensional quantum group tex2html_wrap_inline1137

R. Coquereaux

Next: Introduction

Instituto Balseiro - Centro Atomico de Bariloche

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


Centre de Physique Théorique - CNRS - Luminy, Case 907

F-13288 Marseille Cedex 9 - France


We describe a few properties of the non semi-simple associative algebra tex2html_wrap_inline1137 , where tex2html_wrap_inline1139 is the Grassmann algebra with two generators. We show that tex2html_wrap_inline1123 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 tex2html_wrap_inline1143 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 tex2html_wrap_inline1147 , 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 tex2html_wrap_inline1123 is tex2html_wrap_inline1151 ).

anonymous ftp or gopher:

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

hepth: 9610114


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