Alphabeta MathHall Algebras and Tensor Categories
This collection connects counting and geometric constructions of algebras with tensor categories and their representations. Seven draft theory/example pairs distinguish finitary Ringel–Hall multiplication, Green's compatibility and quantum-group realizations, derived Hall constructions, representations and geometric actions, motivic integration, cohomological Hall correspondences, and stability with Donaldson–Thomas and BPS wall crossing. These constructions have different inputs, coefficient systems, finiteness conditions, and completion requirements. They are not interchangeable definitions of one algebra. Representations of the input category must also be distinguished from modules over its Hall algebra.
Four further pairs address tensor categories, noninvertible symmetry and anyonic chains, modular data and topological field theory, higher categorical defects, and Frobenius sewing in dimension two. Existing categorical traces, braidings, Hopf structures, and Frobenius foundations retain their canonical homes. Any unresolved supplier repair remains a hold before a proposed consumer can use it; a physics interpretation supplies context rather than a mathematical premise.
Readers need the relevant existing group, module, homological, quantum-group, and monoidal-category foundations. The pages give separate reading routes rather than assume that familiarity with one branch supplies every prerequisite of another. All eleven pairs remain draft prose scaffolds. Full local proofs and explicit diagram equations must be supplied before their proposed results can enter an accepted dependency chain.
Track and source contracts: research/plan-physics-algebras-track.md.
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
5 pagesThis page records the geometric construction and conventions for quiver cohomological Hall algebras (CoHAs), separates the ordinary and potential dependent constructions, and…
5 definitions, 6 lemmas, 4 theorems, 1 remarkExamples & counterexamples →- Finitary Ringel–Hall Algebras12 results
This page constructs the untwisted Hall algebra of a small finitary abelian category over an arbitrary commutative coefficient ring.
4 definitions, 5 lemmas, 3 theoremsExamples & counterexamples → This page treats finite-dimensional Frobenius algebra and two-dimensional oriented sewing over an arbitrary field.
3 definitions, 5 lemmas, 2 theoremsExamples & counterexamples →This page constructs the general weighted Reshetikhin–Turaev functor and its full modular-functor factorization from explicit local algebra and marked relative surgery.
4 definitions, 39 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →An anyonic path space is a fixed-boundary Hom space in a finite semisimple fusion category.
4 definitions, 5 lemmas, 2 propositions, 4 theoremsExamples & counterexamples →
Level 1
4 pagesThis page develops Toën's derived Hall algebra over a finite field from a locally finite dg-enhancement.
3 definitions, 7 lemmas, 3 theorems, 1 remarkExamples & counterexamples →The Euler-twisted Hall product uses a supplied small finite-field hereditary category and the subobject convention 0→ N→ R→ M→0.
5 definitions, 25 lemmas, 2 propositions, 4 theorems, 1 corollaryExamples & counterexamples →Hall modules use the stated untwisted Hall multiplication. The generic nilpotent Jordan regular module is transported by the exact normalized Hall–Littlewood map; its Green…
2 definitions, 4 lemmas, 4 theoremsExamples & counterexamples →This page develops the algebraic structure behind higher symmetries and defect composition.
11 definitions, 16 lemmas, 7 theoremsExamples & counterexamples →
Level 2
1 pageFor a finite quiver over a field, this page defines a motivic Hall algebra from quotient stacks of representations and a relative stack-function coefficient theory.
4 definitions, 2 lemmas, 2 theorems, 2 remarksExamples & counterexamples →
Level 3
1 pageThis page proves the finite-field counting branch of Hall wall crossing for every finite acyclic quiver, including wild quivers.
2 definitions, 6 lemmas, 2 theorems, 1 remarkExamples & counterexamples →