Alphabeta MathDivided Powers, Integral Forms, and Quantum Highest Weights
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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
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
Kashiwara's skew derivations eᵢ' and eᵢ'' on the negative half
Definition
The Lusztig divided-power integral form U_A
Definition
Quantum weight modules, weights in P, and integrability
Definition
The contravariant form on an integrable highest weight module
Definition
Quantum Verma modules M(λ), highest weight vectors, and the simple highest weight quotient L(λ)
Definition
The Lusztig-Kashiwara bilinear form on the negative half: the recursion
Lemma
Divided-power multiplication, toral binomials and the Eᵢ⁽ᵃ⁾Fᵢ⁽ᵇ⁾ reordering identity
Lemma
Integral skew-string coefficients on the divided-power negative half
Lemma
Finite-support regularization of quantum label-weight modules
Lemma
The bar involution and the contravariant anti-involution preserve U_A
Lemma
Cartan projection constructs the Verma contravariant form and identifies its radical
Lemma
The dominant quantum Serre quotient is nonzero and integrable
Lemma
Separation of the negative half by iterated skew derivations
Lemma
The unmodified integral torus has a quantum-binomial basis
Lemma
Local q=1 deformation proves integrable highest weight universality
Lemma
Rational-local quantum halves and torus specialize at q=1
Theorem
The negative-half bilinear form is symmetric, nondegenerate and integral
Lemma
The q=1 toral specialization identifies the classical Cartan generators
Theorem
The integrable quantum highest weight module exists and is unique
Proposition
Integrable bounded-above quantum modules with finite-dimensional weight spaces are direct sums of simple highest weight modules
Proposition
Specialization at q=1 recovers the classical enveloping relations
Sources
- 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)
- Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules.'
- 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)
- Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules. The Fusion Category in Action'
- Masaki Kashiwara, On crystal bases of the $q$-analogue of universal enveloping algebras (1991)
- Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21-64
- Masaki Kashiwara, On crystal bases of the q-analogue of the universal enveloping algebra (Duke Math. J. 63 (1991), 465-516)