Приложения квантования в физике и некоммутативных алгебрах» / «Applications of the quantization in Physics and Noncommutative algebras Разавиниа Фаррох

Короткий опис

Разавиниа Фаррох.
Приложения квантования в физике и некоммутативных алгебрах = Applications of the quantization in Physics and Noncommutative algebras : Applications of the quantization in Physics and Noncommutative algebras : диссертация ... кандидата физико-математических наук : 01.01.06 / Farrokh Razavinia; [Место защиты: ФГАОУ ВО «Московский физико-технический институт (национальный исследовательский университет)»]. - Москва, 2021. - 353 с.
Оглавление диссертациикандидат наук Разавиниа Фаррох
Table of Contents
Abstract I
1 Conventions and the organization of the thesis
1.1 Conventions and notation
1.2 Organization of the thesis
2 Introduction
2.1 The Basics: From groups to algebras
2.2 Ore extensions
2.3 Ambiskew polynomial rings
2.4 Generalized Weyl algebras
2.5 Down-up algebras
2.6 Generalized Heisenberg algebras and weak generalized Weyl algebras
2.7 Lie algebras
2.7.1 Adjoints and the Commutator
2.7.2 The Lie algebra sl(2)
2.7.3 Semisimple Lie algebras and the Cartan subalgebra
2.7.4 Root systems
2.7.5 Kac-Moody Lie algebras
2.7.6 Kac-Moody presentation of affine sl2
2.8 Hopf algebras
2.8.1 Algebras
2.8.2 Coalgebras
2.8.3 Morphisms, tensor products, and bialgebras
2.8.4 Antipodes and Hopf algebras
2.8.5 Commutativity, cocommutativity
2.9 Quantum group Uq(sl(2))
2.9.1 q-Analysis and quantum group Uq(sl(2))
2.9.2 Quantum affine algebra Uq(sl2)
2.9.3 Uq (0) and the quantum Serre relations
2.10 Quantization and algebra problems
2.10.1 Free algebras
2.10.2 Matrix representations of algebras
2.10.3 Algebra of generic matrices
2.10.4 The Amitsur-Levitzki theorem
2.10.5 Deformation quantization
2.10.6 Algebraically closed skew field
2.11 Algebras automorphisms and quantization
2.11.1 Jacobian Conjecture
2.11.2 Some results related to the Jacobian Conjecture
2.11.3 Ind-schemes and varieties of automorphisms
2.11.4 Conjecture of Dixmier and quantization
2.11.5 Tame automorphisms
2.11.6 Approximation by tame automorphisms
2.11.7 Holonomic D-modules, Lagrangian submanifolds
2.11.8 Tame automorphisms and the Quantization Conjectures
2.11.9 Quantization of classical algebras
2.12 Torus actions on free associative algebras and the Bialynicki-Birula theorem
3 Local coordinate systems on quantum flag manifolds
3.1 Feigin's homomorphisms on Uq (n)
3.2 The contribution between Quantum Serre relations and screening operators
3.2.1 sl(3) case
3.2.2 affinized Lie algebra sl(2)
3.3 Local integral of motions; Volkov's scheme
3.3.1 Example Uq(sl2); two point invariants
3.3.2 Example Uq (sl2); three point invariants
3.4 Lattice Virasoro algebra
3.4.1 Lattice Virasoro algebra connected to sl2
3.4.2 q-binomial for positive and negative exponent
3.4.3 Formulation for to extend to four and more invariant points134
3.4.4 Generators of lattice Virasoro algebra coming from 2-dimensional representation of sl2
3.4.5 Results; Generators of lattice Virasoro algebra coming from
3 and 4-dimensional representation of sl2
3.5 Conclusion
3.6 Weak Faddeev-Takhtajan-Volkov algebras; Lattice Wn algebras
3.7 Weak Faddeev-Takhtajan-Volkov algebras
3.7.1 Lattice W2 algebra
3.7.2 Lattice W3 algebra
3.7.3 Lattice W4 algebra; main generator
3.7.4 Lattice W5 algebra; main generator
3.7.5 Lattice Wn algebra; main generator
4 Generalized Heisenberg Algebras and their quantum analogs
5 Quantum generalized Heisenberg algebras
5.1 Quantum generalized Heisenberg algebras
5.2 The structure of quantum generalized Heisenberg algebras
5.2.1 Constructing Hq(f, g) as an ambiskew polynomial ring
5.2.2 Constructing Hq(f, g) as a weak generalized Weyl algebra
5.2.3 Some useful equations
5.2.4 Center of quantum generalized Heisenberg algebras
5.2.5 Primality of Hq(f,g)
5.3 Classification of quantum generalized Heisenberg algebras
5.4 The finite-dimensional simple Hq(f, g)-modules
5.4.1 Doubly-infinite weight Hq(f, g)-modules
5.4.2 Finite-dimensional simple Hq(f, g)-modules
5.4.3 Isomorphisms between finite-dimensional simple Hq (f, g)-modules
5.5 Locally finite derivations of Hq(f, g) when deg f >
5.6 Automorphisms of quantum generalized Heisenberg algebras when deg f >
5.7 Gelfand-Kirillov dimension of the quantum generalized Heisenberg algebras
5.8 Hopf quantum generalized Heisenberg algebras
5.8.1 Simple modules over Hopf quantum generalized Heisenberg algebras
5.9 Conclusion/Future Work
5.9.1 Quantum generalized Heisenberg algebras
5.9.2 Generalized Heisenberg algebras
6 Quantization proof of Bergman's centralizer theorem
6.1 Centralizer theorems
6.1.1 Cohn's centralizer theorem
6.1.2 Bergman's centralizer theorem
6.1.3 Centralizer theorem in free group algebras
6.2 Reduction to generic matrix
6.3 Quantization proof of rank one
6.4 Centralizers are integrally closed
6.4.1 Invariant theory of generic matrices
6.4.2 Algebra of generic matrices with traces is integrally closed
6.4.3 Trace algebras
6.4.4 Proof of centralizers are integrally closed
6.4.5 Completion of the proof
6.4.6 On the rationality of subfields of generic matrices
7 Torus actions on free associative algebras, lifting and Bialynicki-Birula type theorems
7.1 Actions of algebraic tori
7.2 Maximal torus action on the free algebra
7.3 Action of F* on F(zi, z2)
7.4 Positive-root torus actions
7.5 Non-linearizable torus actions
7.6 Discussion
Bibliography