Alphabeta MathBalanced Triples and Global Crystal Bases
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- 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
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
Categorification and Geometric Representation Theory
Commutative Algebra
Coxeter Groups
Lie Theory
- Harish Chandra Isomorphism Casimir and Central Characters
- Integrable Highest Weight Kac Moody Modules
- Kac Moody Algebras from Generalized Cartan Matrices
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- The Weyl Kac Character Formula
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
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- 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
Number Theory
Real Analysis
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
2 · Logical flowchart
3 · Items
Lemma
The Lusztig–Kashiwara form, parameter and coproduct dictionary
Definition
The bar lattice and the divided-power integral lattice of a based module
Definition
Balanced triples over the two charts of ℙ¹ and the globalization map
Definition
The integral residue lattices L_A(∞) and L_A(λ)
Definition
The root-height completion for weightwise quasi-R intertwiners
Definition
The Drinfeld pairing between the positive and negative quantum halves
Lemma
The balanced-triple linear algebra over ℙ¹
Lemma
Integral residues of the negative half: freeness, the crystal basis and the form characterization
Lemma
Reversal preserves the negative-half integral and crystal lattices
Lemma
Integrality of the string decomposition and the divided-power string ideals
Lemma
The Drinfeld pairing is nondegenerate in each root degree and matches the shuffle pairing
Lemma
Homogeneous dual-basis sums: basis independence and weightwise finiteness
Lemma
The string-ideal triples are balanced: the induction step (Kashiwara 7.4.1 / JKK 10.4)
Theorem
The highest weight and negative half integral crystal triples are balanced
Theorem
Existence and uniqueness of the weightwise quasi-R intertwiner
Definition
The lower global basis of U_q^- and of a highest weight module
Lemma
The quasi-R intertwiner determines its inverse and the coproduct factorizations
Theorem
Bar invariance, congruence and uniqueness of the lower global basis
Definition
The upper (form-dual) global basis and the required perfectness of the integral pairing
Theorem
The based quotient: projection of the lower global basis to a highest weight module
Theorem
Integral triangular decomposition and freeness of the Lusztig form
Proposition
Specialization of the global bases and the root-of-unity boundary
Theorem
Lusztig's canonical basis and Kashiwara's lower global basis agree in symmetric finite ADE type
Theorem
Finitely generated free integral weight lattices and deformation of the classical weight character
Theorem
Upper and lower global bases are dual and crystal compatible
Lemma
Distinct cyclotomic shears give integrable based quantum sl-two chains
Lemma
Two cyclotomic shears give a periodic integrable based quantum sl-two chain
Lemma
Periodic cyclotomic tensor corrections escape every finite cut
Lemma
Divided-power integrality of the quasi-R action on based lattices
Theorem
Tensor bars on finite products of based integrable modules
Sources
- Lusztig, Introduction to Quantum Groups
- Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras (1991)
- Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516
- Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390
- George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan)
- Huancheng Bao and Weiqiang Wang, Canonical bases in tensor products revisited, arXiv:1403.0039v2
- Jonathan Brundan, Dual Canonical Bases and Kazhdan–Lusztig Polynomials, arXiv:math/0509700v2 (Transform. Groups 14 (2009), 535–558)
- Antoine Lacabanne, Drinfeld Double of Quantum Groups, Tilting Modules and Z-Modular Data Associated to Complex Reflection Groups, arXiv:1807.00770v2
- George Lusztig, Introduction to Quantum Groups, author-hosted complete scan
- Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules. The Fusion Category in Action'
- 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)
- George Lusztig, Introduction to Quantum Groups, based-module conditions and divided-power rank-one calculus
- Huancheng Bao and Weiqiang Wang, Canonical bases in tensor products revisited
- George Lusztig, Introduction to Quantum Groups
- Bao–Wang, Canonical bases in tensor products revisited