Alphabeta MathTensor-Product Crystals and Highest-Weight Crystal Uniqueness
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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
Lemma
Tensor products of integrable weight modules and of their crystal lattices
Definition
The tensor product crystal B₁⊗ B₂ and its two-case convention
Lemma
The i-string components of a lattice vector and the single surviving component modulo q
Definition
Highest weight elements, lowering-generated sets and highest weight crystals
Theorem
The crystal basis of a tensor product and the two-case operator rule
Lemma
The weight, εᵢ and φᵢ formulas for the tensor product crystal
Theorem
The canonical crystal B(λ) is a highest weight crystal of highest weight λ
Proposition
The dual crystal: reversed arrows, negated weights and swapped ε/φ
Theorem
Uniqueness: crystal bases of integrable modules decompose into highest weight crystals
Theorem
Highest weight components in a tensor product crystal
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)
- Mark Shimozono, Crystals for Dummies (lecture notes, 28 August 2005)
- Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516
- Sami Assaf, Anne Dranowski and Nicolle Gonzalez, Extremal tensor products of Demazure crystals, arXiv:2210.10236