Evaluation representations of quantum affine algebras at roots of unity

Abe, Yuuki; Nakashima, Torahiko

doi:10.1063/1.2235047

Cited by 3 publications

(1 citation statement)

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“…We recall some facts about the evaluation representations of the quantum loop algebra U ǫ (Lsl n+1 ), when ǫ is a root of unity. The references are [CP94] (for a generic parameter) and [AN06] (for a root of unity).…”

Section: The Steinberg Tensor Product Theoremmentioning

confidence: 99%

Frobenii on Morava $E$-theoretical quantum groups

Yang,

Zhao

2021

Preprint

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In this paper, we study a family of new quantum groups labelled by a prime number p and a natural number n constructed using the Morava E-theories. We define the quantum Frobenius homomorphisms among these quantum groups. This is a geometric generalization of Lusztig's quantum Frobenius from the quantum groups at a root of unity to the enveloping algebras. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. As an application, we prove a Steinberg-type formula for irreducible representations of these quantum groups. Consequently, we prove that, in type A the characters of certain irreducible representations of these quantum groups satisfy the formulas introduced by Lusztig in 2015. Contents 1. Introduction 2. Topology preliminaries 3. Quiver representations 4. Quiver with automorphism and folding of affine quantum groups 5. The quantum Frobenius maps 6. Discussions and examples 7. Power operations and another construction in the A 1 -case 8. The Frobenius phenomenon on representations 9. The monoidal structure 10. The Steinberg tensor product theorem 11. Characters of irreducible modules Appendix A. The coproduct and Stable envelope References

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Section: The Steinberg Tensor Product Theoremmentioning

confidence: 99%