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A system of generators for tex2html_wrap_inline1123

Let q be a primitive cubic root of unity ( tex2html_wrap_inline1187 ). Hence, tex2html_wrap_inline1299 and tex2html_wrap_inline1301 . We also set tex2html_wrap_inline1303 . In order to write generators for tex2html_wrap_inline1123 , we need to consider tex2html_wrap_inline1267 matrices that have a tex2html_wrap_inline1309 block diagonal structure. We introduce elementary matrices tex2html_wrap_inline1311 for the tex2html_wrap_inline1313 part and elementary matrices tex2html_wrap_inline1315 for the tex2html_wrap_inline1317 part.

The associative algebra tex2html_wrap_inline1123 defined previously can be generated by the following three matrices


Explicitly, one gets




Performing explicit matrix multiplications or using the relations tex2html_wrap_inline1327 , tex2html_wrap_inline1329 and tex2html_wrap_inline1331 , it is easy to see that the following relations are satisfied:


It is relatively easy and straightforward to check that the following tex2html_wrap_inline1333 matrices tex2html_wrap_inline1335 are linearly independent and span tex2html_wrap_inline1123 as vector space over tex2html_wrap_inline1195 . This shows that the matrices tex2html_wrap_inline1341 generate tex2html_wrap_inline1123 as an algebra.

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

Let tex2html_wrap_inline1347 denote the Gell Mann matrices (a basis for the Lie algebra of SU(3)) and tex2html_wrap_inline1351 denote the Pauli matrices (a basis for the Lie algebra of SU(2)). Since we have to use tex2html_wrap_inline1267 matrices, we call tex2html_wrap_inline1357 , tex2html_wrap_inline1359 . Therefore tex2html_wrap_inline1361 , tex2html_wrap_inline1363 and we set tex2html_wrap_inline1365 . We shall also need the SU(2) doubletsgif tex2html_wrap_inline1369 and tex2html_wrap_inline1371 One can then rewrite the generators tex2html_wrap_inline1373 and K as follows:


Before ending this subsection, we want to note that the matrix tex2html_wrap_inline1377 (use tex2html_wrap_inline1379 ) commutes with all elements of tex2html_wrap_inline1123 . If we set tex2html_wrap_inline1383 , tex2html_wrap_inline1385 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 tex2html_wrap_inline1401 associated with eigenvalue -2/3 is isomorphic with the 9 dimensional space tex2html_wrap_inline1407 whereas the eigenspace tex2html_wrap_inline1409 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 tex2html_wrap_inline1417 and tex2html_wrap_inline1419 . We obtain in this way another decomposition of tex2html_wrap_inline1123 as the direct sum of subspaces of dimension 9, 13 and 5 (the supplement).

It is useful to consider the following matrix: tex2html_wrap_inline1429 because its square projects on the block tex2html_wrap_inline1313 of tex2html_wrap_inline1123 . The projector is tex2html_wrap_inline1435 . In the same way, it is useful to consider a matrix that projects on the tex2html_wrap_inline1215 block of tex2html_wrap_inline1123 . One can use tex2html_wrap_inline1441 . 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 tex2html_wrap_inline1443 . These two matrices tex2html_wrap_inline1445 and tex2html_wrap_inline1447 are very useful since they allow us to express any element of tex2html_wrap_inline1123 in terms of the generators tex2html_wrap_inline1373 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 tex2html_wrap_inline1123 , since the coproduct is usually defined on the generators). One can express in this way the 27 elementary matrices (with or without tex2html_wrap_inline1263 's)


For illustration only, we give : tex2html_wrap_inline1461 and tex2html_wrap_inline1463 .

next up previous
Next: The coproduct on Up: No Title Previous: The algebra

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