Iterative construction of the R-matrices in arbitrary dimensions

RESUMO

We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter. By analyzing the spectra of the R-matrices, we conclude that the first and the third series are associated with the symmetric tensor representations of the quantum groups U_q(sl(2)) and U_q(sl(2|1)), respectively. The other two series appear to be related to representations of quantum groups in the case where q is a root of unity. The elements of the second series are associated with nilpotent representations of U_q(sl(2)). We also check the correspondence between these R-matrices and link invariants. Applying the Reshetikhin-Turaev procedure to the first, second, and fourth sequences yields, respectively, colored Jones polynomials, ADO invariants, and Alexander polynomials. To our surprise, a similar calculation for the third series yields a trivial result, identically equal to 1.