Bimodule structure of the mixed tensor product over \(\mathcal{U}_q s \ell(2 | 1)\) and quantum walled Brauer algebra. (English) Zbl 1381.81056
Summary: We study a mixed tensor product \(3^{\otimes m} \otimes \overline{3}^{\otimes n}\) of the three-dimensional fundamental representations of the Hopf algebra \(\mathcal{U}_q s \ell(2 | 1)\), whenever \(q\) is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective \(\mathcal{U}_qs\ell(2|1)\)-module with the generating modules 3 and \(\overline{3}\) are obtained. The centralizer of \(\mathcal{U}_qs\ell(2|1)\) on the mixed tensor product is calculated. It is shown to be the quotient \(\mathsf{X}_{m, n}\) of the quantum walled Brauer algebra \(\mathsf{qw} \mathcal{B}_{m, n}\). The structure of projective modules over \(\mathsf{X}_{m, n}\) is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over \(\mathsf{X}_{m,n}\). This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over \(\mathsf{X}_{m,n} \boxtimes \mathcal{U}_qs\ell(2|1)\). We give an explicit bimodule structure for all \(m,n\).
MSC:
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 82B43 | Percolation |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 16T05 | Hopf algebras and their applications |
| 16D20 | Bimodules in associative algebras |
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