Alphabeta MathQuantum Algebra and Representation Theory
This collection studies algebras through their modules, tensor constructions, and explicit presentations, with three connected branches. The Clifford branch begins with a tensor-algebra quotient and a fixed quadratic-form convention, then proposes basis, classification, graded-product, and Pin and Spin module results. Its comparison with canonical anticommutation relations distinguishes finite algebraic constructions from analytic completions. Spin geometry and real K-theory applications retain their homes in Differential Geometry and Algebraic Topology.
The diagram branch starts with Temperley–Lieb diagrams and a Catalan basis before cellular modules, semisimplicity, specialization, Jones–Wenzl elements, and quotient traces. Brauer, BMW, and partition algebras provide related centralizer problems. A diagram's composition and evaluation must be algebraically defined, and generic, root-of-unity, and positive-characteristic statements have separate contracts. The current and vertex branch connects affine currents and Virasoro operators to state-field identities and module theories, distinguishing rational from logarithmic regimes.
Readers enter through Abstract Algebra, Lie Theory, Representation Theory, and the existing monoidal and Hopf–Hecke foundations. Those categories keep their canonical suppliers; this category supplies proposed extensions and comparison routes. All fourteen theory/example pairs remain draft prose scaffolds with empty item lists. Source evidence, conventions, prerequisite holds, and unfinished local proofs are recorded in the linked track.
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
4 pagesThis page adds the Virasoro/current interface to the existing affine Lie algebra foundation.
7 definitions, 8 lemmas, 1 theoremExamples & counterexamples →This page defines typed Brauer and partition diagrams, a fixed BMW presentation, and their composition rules.
3 definitions, 14 lemmas, 6 theoremsExamples & counterexamples →This page defines the finite Temperley–Lieb diagram algebra over a commutative ring, proves the diagram and generator presentations agree, and identifies the Catalan-sized basis.
2 definitions, 1 lemma, 5 theoremsExamples & counterexamples →For a finite-dimensional vector space V over a field k with char(k)≠2 and symmetric bilinear form B, set Q(v)=B(v,v).
1 definition, 3 lemmas, 2 theoremsExamples & counterexamples →
Level 1
4 pagesThis page fixes the complex CAR relations and the plus-Q Clifford convention.
5 definitions, 6 lemmas, 7 theoremsExamples & counterexamples →This page uses the plus-square convention v̄^ 2=Q(v)1 and orders real signatures as (+ᵖ,-^q).
2 definitions, 7 lemmas, 4 theoremsExamples & counterexamples →For each finite rank n, this page constructs the right link-state modules W n,t over a commutative ring with loop value δ.
1 definition, 3 lemmas, 1 proposition, 2 theorems, 1 corollaryExamples & counterexamples →This page fixes the ordinary, non-super state–field convention Y(a,z)=∑ n∈ℤa (n)z⁻ⁿ⁻¹.
3 definitions, 7 lemmas, 1 theoremExamples & counterexamples →
Level 2
3 pagesThis page fixes the real convention Q(v)=B(v,v) and v²=Q(v)1 for finite-dimensional nondegenerate quadratic spaces of signature (+ᵖ,-^q).
1 definition, 9 lemmas, 4 theorems, 1 corollaryExamples & counterexamples →This page fixes the loop convention δ=q+q⁻¹ and the recurrence for quantum integers before stating any parameter criterion, and distinguishes the complex parameter κ=ord(q²) from the recurrence quantum characteristic.
3 definitions, 8 lemmas, 3 theoremsExamples & counterexamples →This page fixes the module Jacobi identity for the ordinary non-super state–field convention of vertex-algebra-state-field-foundations.
10 definitions, 2 lemmas, 2 theoremsExamples & counterexamples →
Level 3
1 pageThis page keeps three separate subjects: complex root-of-unity standard modules, Spencer’s arbitrary-characteristic decomposition theory, and the negligible quotient/TLJ category.
8 definitions, 21 lemmas, 6 theoremsExamples & counterexamples →
Level 4
2 pagesThis page uses the supplier-first Jones–Wenzl, root-path and full rank-one tilting constructions on temperley-lieb-root-of-unity-and-modular-representation-theory.
1 theoremExamples & counterexamples →This draft develops four distinct interfaces: the abstract A ℓ-1 path-space tower; the full Jones tower of an ordinary finite-index II 1 inclusion; the open XXZ and Potts…
5 definitions, 9 lemmas, 4 theoremsExamples & counterexamples →