Let *q* be a primitive cubic root of unity ( ). Hence,
and . We also set . In order to write generators for , we need to
consider matrices that have a block diagonal structure. We introduce elementary
matrices for the part and elementary matrices
for the part.

The associative algebra defined previously can be generated by the following three matrices

Explicitly, one gets

Performing explicit matrix multiplications or using the relations , and , it is easy to see that the following relations are satisfied:

It is relatively easy and straightforward to check that the following matrices are linearly independent and span as vector space over . This shows that the matrices generate as an algebra.

It is instructive to write these generators in terms of Gell Mann
matrices, Pauli matrices and *SU*(2) doublets :

Let denote the Gell Mann matrices (a
basis for the Lie algebra of *SU*(3)) and
denote the Pauli matrices (a basis for the Lie algebra of *SU*(2)).
Since we have to use matrices, we call , . Therefore
,
and we set
. We shall also
need the *SU*(2) doublets
and
One can then rewrite the generators and *K* as follows:

Before ending this subsection, we want to note that the matrix
(use
)
commutes with all elements of . If we set ,
and let *h* go
to zero (which of course cannot be done when *q* is a root of unity !),
the expression of *C* formally coincides with the usual Casimir operator.

The explicit expression of *C* reads

This operator, when acting by left multiplication on the algebra,
has two eigenvalues (-2/3 and 1/3) and we see explicitly that the
eigenspace associated with eigenvalue -2/3 is isomorphic with the 9
dimensional space whereas the eigenspace associated with eigenvalue 1/3
consists only of nilpotent elements
and coincides with the 13-dimensional radical *J* already
described. In other words, we have and . We obtain in this way another decomposition of as
the direct sum of subspaces of dimension 9, 13 and 5 (the
supplement).

It is useful to consider the following matrix: because its square projects on the block of . The projector is . In the same way, it is useful to consider a matrix that projects on the block of . One can use . This matrix is not a projector but it nevertheless does the required job since it kills the elements of the upper left block. Indeed,

The above properties show that .
These two matrices and are very useful since they allow
us to express any element of in terms of the generators
and *K* (something that is for instance needed, if one wants to
calculate the expression of the coproduct --see below-- for an arbitrary element of
, since the coproduct is usually defined on the generators).
One can express in this way the 27 elementary matrices (with or without 's)

For illustration only, we give : and .

Tue Nov 5 15:18:21 MET 1996