The theory of complex representations of quantum groups at root of unity has been worked out by a number of people. In the case of , see in particular the articles by ,  and . The study of representation theory of the finite dimensional algebra was studied by . Since our attitude, in the present paper, is to study this Hopf algebra without making reference to the general theory of quantum groups, we shall not use this last work but describe the representation theory of by using the explicit definition of the algebra given in the first section.
Since acts on itself (for instance from the left) one may want to consider the problem of decomposition of this representation into irreducible or indecomposable representations (modules). The problem is solved by considering separately all the columns defining as a matrix algebra over a ring. We just ``read'' the following three indecomposable representations from the explicit definition of (the following should be read as ``column vectors''). First of all we have a three dimensional irreducible representation , (where are complex numbers) coming from . Notice that the three columns give equivalent representations. Next we have two reducible indecomposable representations (also called ``PIM's'' for ``Projective Indecomposable Modules'') coming from the columns of . Notice that the first two columns give equivalent representations (that we call ), and the last column gives the representation . Each of these two PIMS is of dimension 6. and . The notation for the three dimensional irreducible representation comes from the fact that, in general algebra, such modules are called ``Steinberg modules''. The PIM's are also called ``principal modules''. Our notation and refers to the fact that, when expressed in terms of Grassmann numbers, and are respectively odd and even.
and , although indecomposable, are not irreducible : submodules (sub-representations) are immediately found by requiring stability of the representation spaces under the left multiplication by elements of .
At first sight (see our modifying comment below) one obtains immediately the following lattice of submodules for the representations and (arrows represent inclusions):
respectively generated by , , , for and by , , , for . Notice that and that . (respectively ) is called the socle of (respectively of ). The module is the radical of and is the radical of .
However, we have forgotten something. Indeed, take , set , define and consider the subspace of spanned by where belong to . This subspace is clearly invariant under the left action of ; moreover two representations corresponding to different values of are inequivalent. Appearance of such inequivalent representations (for different values of ) is related to the fact that the group acts by exterior automorphisms on the algebra , since it ``rotates'' the space spanned by and . Multiplying and by a common scalar multiple amounts to change the coefficient so that this family of representations is indeed parameterized by . The representations and described previously are just two particular members of this family corresponding to the choices and . A similar phenomenon occurs for submodules of the ``odd'' module where we define
The lattices of submodules of and are therefore given by figure 1
Figure 1: The lattices of submodules for the principal modules of
Since we have a totally explicit description of the algebra and of its lattice of representations, it is easy to continue the analysis and to investigate other properties of illustrating many other general concepts from the study of non semi-simple associative algebras. One can, for instance, study the projective covers of the different representations (for completeness sake, this information is represented by dashed lines on figure 1), the subfactor representations, the quiver of the algebra, its Cartan matrix etcThis, however, would be a bit technical and more appropriate for a review paper (see ).
We want only to recall that there exists a one-to-one correspondence between irreducible representations of the algebra and the principal modules , and . Irreducible representations are obtained from these principal modules by factorizing their radical, which amounts to kill the Grassmann `` '' variables. From the above, we see that we obtain in this way three irreducible representations : a representation of dimension 3, (it was already irreducible) which is a triplet for the unitary group U(3) of the part of , a representation of dimension 2, (the quotient of by its radical) which is a doublet for the unitary group U(2) of the part of , and finally a representation of dimension 1, (the quotient of by its radical), a U(1) singlet. These are the three irreducible representations corresponding to the quotient of by its Jacobson radical : (namely ).
The explicit definition given for allows one to compute any tensor products of representations and reduce them. We have to consider the projective indecomposable representations ( , and ) together with the irreducible ones ( , and ). Here again appears a mixing between and via the coproduct, for instance, .