Alphabeta MathQuantum Groups and Crystal Bases
Quantum groups deform enveloping algebras while retaining structures that make highest weights, tensor products and combinatorial bases accessible. This collection starts with quantized enveloping algebras and quantum Serre relations. Divided powers and integral forms prepare quantum highest-weight modules; Kashiwara operators and crystal lattices then describe their combinatorial structure. Tensor-product crystals and highest-weight uniqueness organize how these structures combine. Balanced triples construct global crystal bases with explicit lattice and normalization conventions.
The affine type-A branch develops the level-one Fock-space action and its partition crystal. The residue-crystal page connects these conventions with the symmetric-group applications. Symmetric Groups and Symmetric Functions contains modular branching and its good-node rule; Hecke Algebras and Categorification contains KLR realizations and decomposition-number consequences. Classical root data and Kac–Moody theory remain in Lie Theory, while foundational Hopf-algebra constructions remain in Hopf Algebras & Hecke Algebras. Each topic retains its examples companion and its existing publication status, with prerequisites available through the page links.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Quantum Groups and Crystal Bases
7 pagesQuantized enveloping algebras and divided powers prepare highest-weight modules. Kashiwara operators and crystal lattices lead to tensor-product crystals and global bases. Quantum affine type A supplies the Fock-space action, followed by its residue-crystal conventions for symmetric-group applications. KLR categorification and Hecke decomposition numbers are developed in Hecke Algebras and Categorification.
This page constructs the Drinfeld–Jimbo quantized enveloping algebra of a symmetrizable Cartan datum and develops its quantum Serre relations, its Hopf structure and its triangular decomposition.
7 definitions, 10 lemmas, 6 theoremsExamples & counterexamples →This page develops the integral structure of the Drinfeld–Jimbo algebra and the highest weight representation theory of its integrable modules over the label lattice.
10 definitions, 15 lemmas, 2 propositions, 4 theoremsExamples & counterexamples →- Kashiwara Operators and Crystal Lattices13 results
This page develops the crystal theory of the negative half U q^- and of integrable highest weight modules over the local ring A 0=ℚ[q] (q): the Kashiwara operators extracted…
5 definitions, 4 lemmas, 4 theoremsExamples & counterexamples → This page builds the tensor product of crystals from the crystal lattices of kashiwara-operators-and-crystal-lattices and proves the two structural theorems that make…
2 definitions, 3 lemmas, 1 proposition, 4 theoremsExamples & counterexamples →This page globalizes integral crystal bases at the level of the divided-power form A=ℤ[q,q⁻¹].
7 definitions, 13 lemmas, 1 proposition, 9 theoremsExamples & counterexamples →This page constructs the level-one Fock space of the quantum affine algebra U q(sl e) and identifies its empty connected component with the basic representation: the…
9 definitions, 8 lemmas, 1 proposition, 6 theoremsExamples & counterexamples →This page is the convention bridge between the quantum-group level-one Fock space of quantum-affine-type-a-and-level-one-fock-space and the modular good-node crystal of modular-branching-and-the-good-node-crystal.
2 definitions, 2 lemmas, 1 proposition, 1 theoremExamples & counterexamples →