Evaluation modules for quantum toroidal \(\mathfrak{gl}_n\) algebras. (English) Zbl 1523.17029
Greenstein, Jacob (ed.) et al., Interactions of quantum affine algebras with cluster algebras, current algebras and categorification. In honor of Vyjayanthi Chari on the occasion of her 60th birthday. Based on the summer school and the conference, Washington, DC, USA, June 2018. Cham: Birkhäuser. Prog. Math. 337, 393-425 (2021).
Summary: The affine evaluation map is a surjective homomorphism from the quantum toroidal \(\mathfrak{gl}_n\) algebra \({\mathcal E}^{\prime}_n(q_1,q_2,q_3)\) to the quantum affine algebra \(U^{\prime}_q\widehat{\mathfrak{gl}}_n\) at level \(\kappa\) completed with respect to the homogeneous grading, where \(q_2 = q^2\) and \(q_3^n=\kappa^2\). We discuss \({\mathcal E}^{\prime}_n(q_1,q_2,q_3)\) evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin-type subalgebra of a completion of \({\mathcal E}^{\prime}_n(q_1,q_2,q_3)\), which describes a deformation of the coset theory \(\widehat{\mathfrak{gl}}_n/\widehat{\mathfrak{gl}}_{n-1}\) .
For the entire collection see [Zbl 1481.17001].
For the entire collection see [Zbl 1481.17001].
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
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