Alphabeta MathQuantized Enveloping Algebras and Quantum Serre Relations
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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
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
Bialgebras, counits and antipodes over a commutative ring
Definition
Lie bialgebras, degreewise duality, and root-graded Manin triples
Definition
Symmetrizable Cartan data for quantum groups
Lemma
A nonsingular principal minor of the symmetrized Cartan matrix of size the rank
Definition
Quantum integers, factorials, Gaussian binomials and divided powers at qᵢ
Lemma
An augmented coideal ideal of an enveloping algebra is generated by its primitive part
Lemma
Uniqueness of the antipode
Theorem
A root-graded Manin triple gives dual Lie bialgebras
Definition
The Drinfeld-Jimbo quantized enveloping algebra by generators and relations
Lemma
The opposite Borels of a symmetrizable Kac–Moody algebra are root-degreewise dual Lie bialgebras
Lemma
The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality
Definition
The formal quantum shuffle Borel and its Cartan crossed product
Definition
Positive, negative and toral quantum subalgebras and their root gradings
Lemma
The quantum binomial expansion for q-commuting elements
Lemma
The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal
Lemma
The coproduct preserves the positive and negative quantum Serre ideals
Lemma
The quantum Serre sums vanish in the shuffle algebra, and the opposite Serre ideal annihilates the shuffle half
Lemma
The generic quantum halves form a Drinfeld–Jimbo crossed double
Theorem
The Drinfeld–Jimbo formulas define a Hopf algebra
Theorem
The formal quantum Serre half embeds in the shuffle algebra and is degreewise free
Theorem
Generic quantum Serre halves have classical PBW ranks and a nondegenerate Hopf pairing
Theorem
Triangular decomposition of a quantized enveloping algebra
Theorem
Divided-power commutation and the simple U_qᵢ(sl₂) string modules
Sources
- Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups (lecture notes, arXiv:2106.05252v3)
- 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)
- Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups
- Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390
- Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21–64
- Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I
- A. Kleshchev, Lectures on Infinite Dimensional Lie Algebras
- Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21-64
- K. Conrad, Modules over a PID