On the finite dimensional quantum group M_3 + (M_{2|1}(Lambda^2))_0

Abstract

We describe a few properties of the non semi-simple associative algebra H = M_3 + (M_{2|1}(Lambda2))_0, where Lambda2 is the Grassmann algebra with two generators. We show that H 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 SLq(2) 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 SL(2,F3), 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 H is U(3) x U(2) x U(1)).