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- Title
On the block structure of the quantum ℛ-matrix in the three-strand braids.
- Authors
Bishler, L.; Morozov, An.; Shakirov, Sh.; Sleptsov, A.
- Abstract
Quantum ℛ-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation T of SUq(N) associated with each strand, one needs two matrices: ℛ1 and ℛ2. They are related by the Racah matrices ℛ2=𝒰ℛ1𝒰†. Since we can always choose the basis so that ℛ1 is diagonal, the problem is reduced to evaluation of ℛ2-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that ℛ2-matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of ℛ1 matrix. In this case in order to get a block-diagonal matrix, one should rotate the ℛ2 defined by the Racah matrix in the accidental sector by the angle exactly ±π4.
- Subjects
QUANTUM groups; BRAID theory; MATRICES (Mathematics); RACAH algebra; EIGENVALUES; TOPOLOGICAL fields
- Publication
International Journal of Modern Physics A: Particles & Fields; Gravitation; Cosmology; Nuclear Physics, 2018, Vol 33, Issue 17, pN.PAG
- ISSN
0217-751X
- Publication type
Article
- DOI
10.1142/S0217751X18501051