Quantum Sugawara operators in type \(A\).(English)Zbl 1551.17016
Summary: The quantum Sugawara operators associated with a simple Lie algebra \(\mathfrak{g}\) are elements of the center of a completion of the quantum affine algebra \(\mathrm{U}_q (\hat{\mathfrak{g}})\) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series \(\ell_V(z)\) associated with every finite-dimensional representation \(V\) of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over \(\mathrm{U}_q (\hat{\mathfrak{g}})\) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the \(q\)-characters and \(q\)-deformed \(\mathcal{W}\)-algebras via the Harish-Chandra homomorphism.
We produce explicit quantum Sugawara operators for the quantum affine algebra of type \(A\) which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the \(q\)-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.
We produce explicit quantum Sugawara operators for the quantum affine algebra of type \(A\) which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the \(q\)-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20C08 | Hecke algebras and their representations |
Cite
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