| Description |
1 online resource (140 p.) |
| Series |
SpringerBriefs in mathematical physics ; volume 49 |
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SpringerBriefs in mathematical physics ; v. 49.
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| Contents |
Intro -- Preface -- Acknowledgments -- Contents -- 1 Quantum Loop mathfraksln, Its Two Integral Forms, and Generalizations -- 1.1 Algebras Uv(Lmathfraksln),mathfrakUv(Lmathfraksln),Uv(Lmathfraksln) and Their Bases -- 1.1.1 Quantum Loop Algebra Uv(Lmathfraksln) and Its PBWD Bases -- 1.1.2 Integral Form mathfrakUv(Lmathfraksln) and Its PBWD Bases -- 1.1.3 Integral Form Uv(Lmathfraksln) and Its PBWD Bases -- 1.2 Shuffle Realizations and Applications to the PBWD Bases -- 1.2.1 Shuffle Algebra S(n) -- 1.2.2 Proofs of Theorems 1.1 and 1.8 |
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1.2.3 Integral Form mathfrakS(n) and a Proof of Theorem 1.3 -- 1.2.4 Integral Form S(n) and a Proof of Theorem 1.6 -- 1.3 Generalizations to Two-Parameter Quantum Loop Algebras -- 1.3.1 Quantum Loop Algebra U>v1,v2(Lmathfraksln) and Its PBWD Bases -- 1.3.2 Integral Form mathfrakU>v1,v2(Lmathfraksln) and Its PBWD Bases -- 1.3.3 Shuffle Algebra widetildeS(n) -- 1.3.4 Proofs of Theorems 1.16 and 1.18 -- 1.4 Generalizations to Quantum Loop Superalgebras -- 1.4.1 Quantum Loop Superalgebra U>v(Lmathfraksl(m )) -- 1.4.2 PBWD Bases of U>v(Lmathfraksl(m )) |
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1.4.3 Integral Form mathfrakU>v(Lmathfraksl(m )) and Its PBWD Bases -- 1.4.4 Shuffle Algebra S(m ) -- 1.4.5 Proofs of Theorems 1.19 and 1.21 -- References -- 2 Quantum Toroidal mathfrakgl1, Its Representations, and Geometric Realization -- 2.1 Quantum Toroidal Algebras of mathfrakgl1 -- 2.1.1 Quantum Toroidal mathfrakgl1 -- 2.1.2 Elliptic Hall Algebra -- 2.1.3 ̀̀90 Degree Rotation'' Automorphism -- 2.1.4 Shuffle Algebra Realization -- 2.1.5 Neguţ's Proof of Theorem 2.3 -- 2.1.6 Commutative Subalgebra mathcalA -- 2.2 Representations of Quantum Toroidal mathfrakgl1 -- 2.2.1 Categories mathcalOpm |
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2.2.2 Vector, Fock, Macmahon Modules, and Their Tensor Products -- 2.2.3 Vertex Representations and Their Relation to Fock Modules -- 2.2.4 Shuffle Bimodules and Their Relation to Fock Modules -- 2.3 Geometric Realizations -- 2.3.1 Correspondences and Fixed Points for (mathbbA2)[n] -- 2.3.2 Geometric Action I -- 2.3.3 Heisenberg Algebra Action on the Equivariant K-Theory -- 2.3.4 Correspondences and Fixed Points for M(r,n) -- 2.3.5 Geometric Action II -- 2.3.6 Whittaker Vector -- References -- 3 Quantum Toroidal mathfraksln, Its Representations, and Bethe Subalgebras |
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3.1 Quantum Toroidal Algebras of mathfraksln -- 3.1.1 Quantum Toroidal mathfraksln (n3) -- 3.1.2 Hopf Pairing, Drinfeld Double, and a Universal R-Matrix -- 3.1.3 Horizontal and Vertical Copies of Quantum Affine mathfraksln -- 3.1.4 Miki's Isomorphism -- 3.1.5 Horizontal and Vertical Extra Heisenberg Subalgebras -- 3.2 Shuffle Algebra and Its Commutative Subalgebras -- 3.2.1 Shuffle Algebra Realization -- 3.2.2 Commutative Subalgebras mathcalA(s0,...,sn-1) -- 3.2.3 Proof of Theorem 3.7 -- 3.2.4 Special Limit mathcalAh -- 3.2.5 Identification of Two Extra Horizontal Heisenbergs |
| Summary |
This book is based on the author's mini course delivered at Tokyo University of Marine Science and Technology in March 2019. The shuffle approach to DrinfeldJimbo 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 FeiginOdesskii 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 |
| Notes |
Description based upon print version of record |
| Bibliography |
Includes bibliographical references |
| Notes |
3.3 Representations of Quantum Toroidal mathfraksln |
| Subject |
Quantum groups.
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Quantum groups.
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| Form |
Electronic book
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| ISBN |
9789819931507 |
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9819931509 |
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