Overview
- Shuffle approach is a powerful technique in treating both algebraic and geometric aspects of quantum affinized algebras
- Collects in one volume information about shuffle algebras which usually is spread over various papers from two decades
- Highlights key algebraic aspects of shuffle algebras and is intended for a wide math and physics audience
Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 49)
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Table of contents (3 chapters)
Keywords
About this book
The shuffle approach to Drinfeld–Jimbo quantum groups of finite type (embedding their "positive" subalgebras into q-deformed shuffle algebras) was first developed independently in the 1990s by J. Green, M. Rosso, and P. Schauenburg. Motivated by similar ideas, B. Feigin and A. Odesskii proposed a shuffle approach to elliptic quantum groups around the same time. The shuffle algebras in the present book can be viewed as trigonometric degenerations of the Feigin–Odesskii elliptic shuffle algebras. They provide combinatorial models for the "positive" subalgebras of quantum affine algebras in their loop realizations. These algebras appeared first in that context in the work of B. Enriquez.
Over the last decade, the shuffle approach has been applied to various problems in combinatorics (combinatorics of Macdonald polynomials and Dyck paths, generalization to wreath Macdonald polynomials and operators), geometric representation theory (especially the study of quantum algebras’ actions on the equivariant K-theories of various moduli spaces such as affine Laumon spaces, Nakajima quiver varieties, nested Hilbert schemes), and mathematical physics (the Bethe ansatz, quantum Q-systems, and quantized Coulomb branches of quiver gauge theories, to name just a few).
While this area is still under active investigation, the present book focuses on quantum affine/toroidal algebras of type A and their shuffle realization, which have already illustrated a broad spectrum of techniques. The basic results and structures discussed in the book are of crucial importance for studying intrinsic properties of quantum affinized algebras and are instrumental to the aforementioned applications.
Authors and Affiliations
Bibliographic Information
Book Title: Shuffle Approach Towards Quantum Affine and Toroidal Algebras
Authors: Alexander Tsymbaliuk
Series Title: SpringerBriefs in Mathematical Physics
DOI: https://doi.org/10.1007/978-981-99-3150-7
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023
Softcover ISBN: 978-981-99-3149-1Published: 08 August 2023
eBook ISBN: 978-981-99-3150-7Published: 07 August 2023
Series ISSN: 2197-1757
Series E-ISSN: 2197-1765
Edition Number: 1
Number of Pages: XI, 130
Number of Illustrations: 1 b/w illustrations
Topics: Mathematical Physics, General Algebraic Systems, Topological Groups, Lie Groups