Alphabeta MathHopf Algebras & Hecke Algebras
Hopf algebras describe symmetries that act coherently on tensor products and finite dual representations. Hecke algebras deform reflection-group relations and organize families of representations with controlled bases and traces. This collection develops both theories from their elementary algebraic constructions, beginning with tensor descent, quotient universal properties, finite duality and coherence. Coalgebras and comodules lead to bialgebras, convolution and antipodes; quotient and finite-dual constructions then lead to Hopf modules, integrals, Frobenius forms, quasitriangular structures and finite quantum doubles. The Hecke branch proves Coxeter exchange and reduced word control before introducing its generic basis, parabolic induction, symmetrizing trace, specialization, affine and cyclotomic constructions, and cellular module theory. A concrete Hecke symmetry connects the branches through a tensor-power action; generic quantum centralizers have their own separate proof obligations. Every definition's existence, descent, uniqueness or compatibility is assigned an explicit local supplier contract.
These pages are prose scaffolds with empty item lists, awaiting full local
proof authoring and review. Existing application homes are retained:
principal-series-representations-of-gl-n-over-a-finite-field,
affine-group-schemes-hopf-algebras-and-rational-representations, and
type-a-soergel-bimodules-and-hecke-categorification. Braid Groups remains
an independent category, and Special Topics in Representation Theory retains
its specialized representation tracks. Source evidence and exact prerequisite
contracts live in research/plan-hopf-hecke-algebras-track.md.
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 · Tensor and Coalgebra Foundations
5 pagesEstablish tensor descent and coherence before coalgebras, finite coefficient constructions, comodules, convolution and antipodes. Prove every quotient and finite-dual formula is well-defined before using it.
The diagonal action of a group on two representations uses g↦g⊗g, while a primitive symmetry uses x↦x⊗1+1⊗x.
A coalgebra reverses the structure arrows of an associative algebra: one input is split into two outputs.
A right comodule records a vector together with the coalgebra coefficients of its transformation.
A quotient Hopf algebra requires three different descents: multiplication, coproduct/counit, and antipode.
- Tensor Coherence and Algebraic Descent10 results
The page fixes the tensor conventions shared by the Hopf and Hecke branches: k is a field, ⊗ means ⊗ k over k-vector spaces, tensor powers are left-associated with the empty…
1 definition, 9 lemmasExamples & counterexamples →
Part 2 · Hopf Representations and Finite Structures
5 pagesConstruct tensor and finite dual representations, prove the Hopf-module inverse, then derive integrals and finite antipode bijectivity. Quasitriangular structures and finite doubles build on these suppliers.
The quantum double combines a finite Hopf algebra and its dual into an algebra with controlled cross-relations.
An integral is a vector in the regular module transforming by the trivial character.
Tensor representations explain why a bialgebra has a coproduct and counit; finite dual representations explain the antipode.
A Hopf module carries both an action and a coaction with an exact compatibility equation.
An ordinary flip generally fails to intertwine a noncocommutative tensor action.
Part 3 · Coxeter and Hecke Foundations
3 pages · after Part 1Prove Coxeter exchange and reduced-word independence before the generic Hecke basis. Parabolic freeness, trace perfection and carefully qualified specialization follow in dependency order.
Deformation arguments use hypotheses on the coefficient ring, field and parameter.
Parabolic induction and traces expose useful structure without yet invoking semisimplicity.
A generic Hecke algebra replaces each involution relation by a quadratic relation while retaining braid relations.
1 definition, 3 lemmas, 1 theoremExamples & counterexamples →
Part 4 · Affine, Cyclotomic, and Tensor Constructions
4 pagesProve affine Laurent divisibility and PBW, then cyclotomic spanning and independence before integral cellular structure. The concrete tensor operator gives a Hecke action; full generic quantum centralizers remain a separately gated proof target.
Cyclotomic Hecke algebras quotient the affine algebra by a polynomial in X1.
Cellular structure organizes representations at singular parameters, where generic eigenvector formulas no longer apply.
The bridge starts with an explicit operator rather than an assumed universal R-matrix.
Affine type A adds commuting invertible weight variables to the finite Hecke generators.