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The representation theory of tex2html_wrap_inline1123

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 tex2html_wrap_inline1615 , see in particular the articles by [29], [2] and [17]. The study of representation theory of the finite dimensional algebra tex2html_wrap_inline1617 was studied by [30]. 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 tex2html_wrap_inline1123 by using the explicit definition of the algebra given in the first section.

Since tex2html_wrap_inline1123 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 tex2html_wrap_inline1623 as a matrix algebra over a ring. We just ``read'' the following three indecomposable representations from the explicit definition of tex2html_wrap_inline1123 (the following should be read as ``column vectors''). First of all we have a three dimensional irreducible representation tex2html_wrap_inline1627 , (where tex2html_wrap_inline1629 are complex numbers) coming from tex2html_wrap_inline1169 . 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 tex2html_wrap_inline1173 . Notice that the first two columns give equivalent representations (that we call tex2html_wrap_inline1635 ), and the last column gives the representation tex2html_wrap_inline1637 . Each of these two PIMS is of dimension 6. tex2html_wrap_inline1641 and tex2html_wrap_inline1643 . The notation tex2html_wrap_inline1645 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 tex2html_wrap_inline1637 and tex2html_wrap_inline1635 refers to the fact that, when expressed in terms of Grassmann numbers, tex2html_wrap_inline1637 and tex2html_wrap_inline1635 are respectively odd and even.

tex2html_wrap_inline1637 and tex2html_wrap_inline1635 , 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 tex2html_wrap_inline1123 .

At first sight (see our modifying comment below) one obtains immediately the following lattice of submodules for the representations tex2html_wrap_inline1637 and tex2html_wrap_inline1635 (arrows represent inclusions):



respectively generated by tex2html_wrap_inline1669 , tex2html_wrap_inline1671 , tex2html_wrap_inline1673 , tex2html_wrap_inline1675 for tex2html_wrap_inline1637 and by tex2html_wrap_inline1679 , tex2html_wrap_inline1681 , tex2html_wrap_inline1683 , tex2html_wrap_inline1685 for tex2html_wrap_inline1635 . Notice that tex2html_wrap_inline1689 and that tex2html_wrap_inline1691 . tex2html_wrap_inline1693 (respectively tex2html_wrap_inline1695 ) is called the socle of tex2html_wrap_inline1637 (respectively of tex2html_wrap_inline1635 ). The module tex2html_wrap_inline1701 is the radical of tex2html_wrap_inline1637 and tex2html_wrap_inline1705 is the radical of tex2html_wrap_inline1635 .

However, we have forgotten something. Indeed, take tex2html_wrap_inline1709 , set tex2html_wrap_inline1711 , define tex2html_wrap_inline1713 and consider the subspace tex2html_wrap_inline1715 of tex2html_wrap_inline1635 spanned by tex2html_wrap_inline1719 where tex2html_wrap_inline1721 belong to tex2html_wrap_inline1195 . This subspace is clearly invariant under the left action of tex2html_wrap_inline1123 ; moreover two representations corresponding to different values of tex2html_wrap_inline1727 are inequivalent. Appearance of such inequivalent representations (for different values of tex2html_wrap_inline1727 ) is related to the fact that the group tex2html_wrap_inline1731 acts by exterior automorphisms on the algebra tex2html_wrap_inline1123 , since it ``rotates'' the space spanned by tex2html_wrap_inline1207 and tex2html_wrap_inline1209 . Multiplying tex2html_wrap_inline1739 and tex2html_wrap_inline1741 by a common scalar multiple amounts to change the coefficient tex2html_wrap_inline1743 so that this family of representations is indeed parameterized by tex2html_wrap_inline1711 . The representations tex2html_wrap_inline1747 and tex2html_wrap_inline1749 described previously are just two particular members of this family corresponding to the choices tex2html_wrap_inline1751 and tex2html_wrap_inline1753 . A similar phenomenon occurs for submodules of the ``odd'' module tex2html_wrap_inline1637 where we define tex2html_wrap_inline1757

The lattices of submodules of tex2html_wrap_inline1637 and tex2html_wrap_inline1635 are therefore given by figure 1

Figure 1: The lattices of submodules for the principal modules of tex2html_wrap_inline1123

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 tex2html_wrap_inline1123 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 [10]).

We want only to recall that there exists a one-to-one correspondence between irreducible representations of the algebra tex2html_wrap_inline1123 and the principal modules tex2html_wrap_inline1645 , tex2html_wrap_inline1635 and tex2html_wrap_inline1637 . Irreducible representations are obtained from these principal modules by factorizing their radical, which amounts to kill the Grassmann `` tex2html_wrap_inline1263 '' variables. From the above, we see that we obtain in this way three irreducible representations : a representation of dimension 3, tex2html_wrap_inline1645 (it was already irreducible) which is a triplet for the unitary group U(3) of the tex2html_wrap_inline1169 part of tex2html_wrap_inline1123 , a representation of dimension 2, tex2html_wrap_inline1791 (the quotient of tex2html_wrap_inline1635 by its radical) which is a doublet for the unitary group U(2) of the tex2html_wrap_inline1215 part of tex2html_wrap_inline1123 , and finally a representation of dimension 1, tex2html_wrap_inline1803 (the quotient of tex2html_wrap_inline1637 by its radical), a U(1) singlet. These are the three irreducible representations corresponding to the quotient tex2html_wrap_inline1275 of tex2html_wrap_inline1123 by its Jacobson radical : (namely tex2html_wrap_inline1813 ).

The explicit definition given for tex2html_wrap_inline1123 allows one to compute any tensor products of representations and reduce them. We have to consider the projective indecomposable representations ( tex2html_wrap_inline1817 , tex2html_wrap_inline1819 and tex2html_wrap_inline1821 ) together with the irreducible ones ( tex2html_wrap_inline1823 , tex2html_wrap_inline1825 and tex2html_wrap_inline1821 ). Here again appears a mixing between tex2html_wrap_inline1169 and tex2html_wrap_inline1831 via the coproduct, for instance, tex2html_wrap_inline1833 .

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Next: Others avatars for and Up: No Title Previous: The coproduct on

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