دانلود کتاب Shuffle Approach Towards Quantum Affine and Toroidal Algebras

فهرست مطالب :

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
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|n))
1.4.2 PBWD Bases of U>v(Lmathfraksl(m|n))
1.4.3 Integral Form mathfrakU>v(Lmathfraksl(m|n)) and Its PBWD Bases
1.4.4 Shuffle Algebra S(m|n)
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
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
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
3.3 Representations of Quantum Toroidal mathfraksln
3.3.1 Vector, Fock, and Macmahon Modules
3.3.2 Vertex Representations
3.3.3 Shuffle Bimodules
3.4 Identification of Representations
3.4.1 Shuffle Realization of Fock Modules
3.4.2 Generalization for Tensor Products of Fock and Macmahon
3.4.3 Twisting Vertex Representations by Miki\'s Isomorphism
3.5 Bethe Algebra Realization of mathcalA(s0,…,sn-1)
3.5.1 Trace Functionals
3.5.2 Functionals via Pairing
3.5.3 Transfer Matrices and Bethe Subalgebras
3.5.4 Bethe Algebra Incarnation of mathcalA(s0,…,sn-1)
3.5.5 Shuffle Formulas for Miki\'s Twist of Cartan Generators
References