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 .