Alphabeta MathKashiwara Operators and Crystal Lattices
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
Braid Groups
Categorification and Geometric Representation Theory
Commutative Algebra
Coxeter Groups
Lie Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Definition
The local ring A₀ of rational functions regular at q=0
Theorem
Uniqueness of the i-string decomposition of an integrable module
Definition
The Kashiwara operators ẽᵢ and f̃ᵢ of an integrable module
Theorem
The i-string decomposition of the negative half
Definition
Crystal lattices, crystal bases and their crystals
Lemma
Rank-one tensor strings and their crystal limit rule
Definition
The crystal graph, εᵢ, φᵢ and the weight function
Definition
The Kashiwara lattices L(λ) and L(∞) and their generating sets
Lemma
Highest weight tensor maps, their adjunction and the retraction S_λ,μ
Theorem
The grand loop: existence of the highest weight crystal bases
Lemma
Crystal lattice polarizations are perfect at q=0
Lemma
Projection from L(∞) to L(λ) and its compatibility with the Kashiwara operators
Theorem
The crystal basis B(∞) of the negative half
Sources
- Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390
- Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024)
- Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516