Alphabeta MathQuantized Enveloping Algebras and Quantum Serre Relations — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
Coxeter Groups
Lie Theory
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Counterexample
Unsymmetrized parameters break the coproduct of the Serre ideal
Example
The quasiprimitive Serre element in type A₂
Example
The double-edge Serre relation for the cyclic affine type A₁⁽¹⁾
Example
Coproduct, antipode and q-binomial expansion in U_q(sl₂)
Sources
- Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21-64
- 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
- Alexander Kleshchev, Lectures on Infinite Dimensional Lie Algebras
- 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)
- Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups (lecture notes, CMSA Math Science Literature Lecture Series, 2020)