Introduction

Quantum groups - either specialized at roots of unity or not - have been used many times, during the last decade, in the physics of integrable models and in conformal theories [29]. The wish of using such mathematical structures, both nice and new, in four-dimensional particle physics has triggered the imagination of many people in the last years.

When q is a root of unity, there are other interesting objects besides the quantized enveloping algebra itself : some of its finite dimensional subalgebras or some of its finite dimensional quotients may still carry a Hopf algebra structure.

We do not need to mention the importance of finite dimensional classical symmetries in Physics, but it is our belief that finite dimensional quantum symmetries will turn out to be also of prime importance in theories of fundamental interactions -in this respect, one can already mention the relations between quantum symmetries of graphs [27] and the classification of conformal field theories [14].

Such finite dimensional quantum groups are also interesting from the mathematical point of view because they provide examples of finite dimensional Hopf algebras which are neither commutative nor co-commutative : they are a kind of direct ``quantum'' generalization of discrete groups (or, better, generalizations of the corresponding group algebras). These objects are also interesting because of their --still not totally understood-- relations with the theory of modular representations of algebraic groups [20][21].

The fact that the semi-simple part of a finite dimensional quotient of the quantum algebra , when q is a primitive cubic root of unity, has a unitary group equal to suggests that this finite quantum group could have something to do with the Standard Model of particle physics. This remark was explicitly made in the framework of non-commutative geometry by Connes in [4] and more recently in [6]. We do not intend, in the present paper, to show how to analyze this finite quantum group along the lines of non-commutative geometry (for a very detailed account of the Standard Model ``à la Connes'', not involving quantum groups at all, we refer to the recent papers [25] or [6]).

In order to make use of a symmetry in physics, it is good to be already acquainted with it. It could be tempting to assume that the reader knows already everything about representation theory of non semi-simple algebras, Jacobson radical, quivers, Hopf algebras and other niceties belonging to the toolbox of the perfect algebraist but this would amount to assume that nobody can appreciate the beauty of a tetrahedron before being acquainted with the properties of the exceptional Lie group . Our point of view is that, since the properties of the algebra can be understood without using anything more sophisticated than basic multiplication or tensor products of matrices as well as elementary calculus involving anti-commuting numbers (Grassmann numbers), it is very useful to study them in this way, at least in a first approach.

We therefore want to present --in very simple terms-- the rather nice finite dimensional algebra of quantum symmetries mentioned before, without assuming from the reader any a priori knowledge on quantum groups, general associative algebras and the like. We shall therefore define explicitly this finite dimensional quantum group, as the algebra , where is the set of matrices over the complex numbers, and where is the Grassmann envelope of the associative graded algebra , i.e. the even part of its graded tensor product with a Grassmann algebra with two generators.

The motivation and underlying belief is, of course, that there should be some quantum symmetry, hitherto unnoticed, in the Standard Model, or, maybe, in a modification of it, symmetry that would, ultimately, cast some light on the puzzle of fermionic families and mass matrices.

The present paper is not only a pedagogical exercise: although several properties that we shall describe have been already discussed in the literature (see in particular [1],[30]), usually using a less elementary language, others do not seem to be published. Sections 2 to 5 are supposed to be elementary and self-contained; the last two sections contain a set of less elementary results and unrelated comments.

Finite dimensional quantum groups associated with quantum universal enveloping algebras can be defined for any type of Lie group, when the parameter q is a primitive root of unity. It is possible to give an explicit realization -- in terms of matrices with complex and grassmanian entries -- for the other finite quantum groups of SL(2) type when q is a root of the unity (see [10], [28]). Following hopes or claims that such algebras can provide interesting physical models, it seems that there is some need for a paper explaining the basic properties of the simplest non trival quantum group of this type, namely, when . This is the purpose of the present paper. Construction of a ``generalized'' gauge theory on , along the lines of non commutative geometry, is clearly possible but lies beyond the scope of this article.