Твистованные представления тороидальных алгебр и их применения Гонин Роман Романович

Короткий опис

Гонин, Роман Романович.
Твистованные представления тороидальных алгебр и их применения = Twisted representations of toroidal algebras and their applications : Twisted representations of toroidal algebras and their applications : диссертация ... кандидата физико-математических наук : 01.01.06 / Роман Гонин; [Место защиты: ФГАОУ ВО «Национальный исследовательский университет «Высшая школа экономики»]. - Москва, 2021. - 137 с. разд.паг.
Оглавление диссертациикандидат наук Гонин Роман Романович
Contents
Introduction
1 Schur specialization case
1.1 Introduction
1.2 q-difference operators
1.2.1 SL2(Z) action
1.2.2 Chevalley generators and relations
1.3 Fock module
1.3.1 Free boson realization
1.3.2 Free fermion realization
1.3.3 Semi-Infinite construction
1.4 Explicit formulas for twisted representation
1.4.1 Fermionic construction
1.4.2 Bosonic construction
1.4.3 Strange Bosonic construction
1.5 Twisted representation via a sublattice
1.5.1 Sublattices and subalgebras
1.5.2 Twisted Fock vs restricted Fock
1.5.3 Explicit formulas for restricted Fock
1.6 Twisted representation via a Semi-infinite construction
1.7 q-W -Algebras
1.7.1 Definitions
1.7.2 Connection of Wq(sln,ntw) with Diffq
1.7.3 Bosonization of Wq(s[n,ntw)
1.7.4 Verma modules vs Fock modules
1.8 Restriction on Diffq for general sublattice
1.8.1 Decomposition of restriction
1.8.2 Strange Bosonization and Odd Bosonization
1.9 Relations on conformal blocks
1.9.1 Whittaker vector
1.9.2 Shapovalov form
1.9.3 Conformal blocks
1.10 Regular product
1.11 Serre relation
1.11.1 Operator product expansion for E(w) ■ ■ ■ E(wk)
1.12 Homomorphism from Diffq to W-algebra
1.13 Whittaker vector
1.13.1 Uniqueness of Whittaker vector
1.13.2 Construction of Whittaker vector
1.13.3 Whittaker vector for Wq(sin) algebra
2 Twisted and non-twisted Virasoro
2.1 Introduction
2.2 Bosonization of Uq(sl2) and its vertex operators
2.3 Vertex operators relations revisited
2.3.1 R-matrix relations
2.3.2 Special point relation
2.3.3 Interchanging relation on Ф and Ф
2.4 Realization of (Twisted) Deformed Virasoro algebra
2.4.1 Deformed Virasoro algebra
2.4.2 Twisted Deformed Virasoro algebra
3 Semi-infinite construction
3.1 Double Affine Hecke Algebra
3.1.1 Cherednik representation
3.2 Representation
3.2.1 Explicit construction
3.2.2 Triangularity of Macdonald operators
3.2.3 Monomial basis
3.2.4 Twisted Cherednik representation
3.3 Toroidal algebra
3.4 Deformed exterior power
3.4.1 Finite v-wedge
3.4.2 The limit
3.5 Semi-infinite construction of twisted Fock module I
3.5.1 Finite case
3.5.2 The limit for the right and left halves
3.5.3 Analytic continuation
3.5.4 Example n =
3.6 Semi-infinite construction of twisted Fock module II
3.6.1 The limit for the bottom half
3.6.2 Action of the whole algebra
3.7 Standard basis
3.8 Quantum affine algebra and its vertex operators
3.8.1 Action of quantum affine algebra
3.8.2 Vertex operators
3.8.3 Factorization of the vertex operator
Conclusion
Bibliography