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Hopf, Hecke, Quantum-Group, and Braid Interfaces

1 · Prerequisites

2 · Summary

This page coordinates the existing Hopf, Hecke, quantum-group, KLR, and braid routes. The finite algebraic R-matrix, generic quantum Schur–Weyl theorem, and invertible Yang–Baxter-to-braid theorem remain at their canonical homes. The local addition is the explicitly bounded rational spectral R-matrix on $C^{2}$ , with its complete parameterized Yang–Baxter proof.

Companion: hopf-hecke-and-quantum-symmetry-interfaces-examples.

Reading prerequisites

The current plan orders this page after algebraic-representation-and-tensor-symmetry-map, hecke-symmetries-and-generic-quantum-schur-weyl-duality, yang-baxter-operators-and-quantum-braid-representations, and klr-algebras-induction-and-quantum-serre-categorification. Page order is reading context; each item's logical dependencies are listed separately in its own carrier.

Existing library route

quasitriangular-hopf-algebras-and-braided-module-categories: finite algebraic R-matrix and module braiding. The universal R is in the algebraic tensor square $H \otimes H$ ; a completed $U_{q}$ series requires a distinct action proof.
generic-coxeter-hecke-algebras-and-the-standard-basis: generic Hecke basis. Specialization conditions remain explicit.
hecke-symmetries-and-generic-quantum-schur-weyl-duality: matrix-image centralizers over $Q (q)$ . No positive-characteristic or root-of-unity conclusion.
yang-baxter-operators-and-quantum-braid-representations: Yang–Baxter operator to braid representation. Operator invertibility is required.
klr-algebras-induction-and-quantum-serre-categorification: KLR decategorification with its exact quantum parameter and grading.

Scope and local proof obligations

Existing interfaces

A quasitriangular structure gives and is recovered from a module braiding is the algebraic finite-tensor $R \in H \otimes H$ convention for all left $H$ -modules. It does not supply a completed universal $R$ .
The ordered-basis symmetry gives the normalized Hecke tensor action and Generic quantum Schur–Weyl mutual centralizers over Q(q) retain the ordered-basis type-A Hecke action and its generic $Q (q)$ image centralizers. The latter does not assert a root-of-unity or positive-characteristic centralizer theorem; its abstract Hecke action is faithful only when $N \geq n$ .
A Yang–Baxter operator gives braid-group representations requires an invertible Yang–Baxter operator. The new spectral family is parameterized, and its singular parameter values are worked out on the B page.
The KLR/categorification route remains at klr-algebras-induction-and-quantum-serre-categorification, with its own coefficient and grading conventions.

Parameter map

The generic Coxeter convention uses $(T_{s} - v_{s}) (T_{s} + v_{s}^{- 1}) = 0$ . The local quantum tensor operator uses $(T_{i} - q) (T_{i} + q^{- 1}) = 0$ . A multiplicative generator $S_{i} = q T_{i}$ instead satisfies $(S_{i} - q^{2}) (S_{i} + 1) = 0$ , so the common type-A parameter is $ξ = q^{2}$ . These are rescalings of generators and parameters, not permission to transfer a semisimplicity or centralizer theorem through specialization.

New local spectral interface

The rational spin-1/2 spectral R-matrix fixes the two-dimensional rational family $R (u) = u I + i P$ and its checked operator. The rational spin-1/2 spectral R-matrix satisfies Yang-Baxter proves both the un-checked parameterized Yang–Baxter equation and its checked braid-form identity by a direct $S_{3}$ operator calculation. The item concerns finite-dimensional vector spaces; it is not a universal Hopf $R$ -matrix or a braiding on a category of affine modules. Full RTT, arbitrary spectral transfer matrices, and all-type evaluation-module claims remain outside this local result.

Canonical suppliers and reuse boundaries

thm-hh-r-matrix-constructs-a-module-braiding: finite algebraic $R$ and all left modules. Not a completed universal $R$ .
thm-hh-generic-quantum-schur-weyl-mutual-centralizers: generic $Q (q)$ mutual centralizers. The matrix-image equalities hold for all $N \geq 1, n \geq 0$ over $Q (q)$ ; abstract Hecke faithfulness is only claimed for $N \geq n$ . No root-of-unity or positive-characteristic extension.
thm-a-yang-baxter-operator-gives-braid-group-representations: an invertible Yang–Baxter operator yields a braid action. Yang–Baxter without invertibility is insufficient.
def-hh-rational-spectral-r-matrix and lem-hh-rational-spectral-r-matrix-satisfies-yang-baxter: finite-dimensional rational spectral family and its fully proved difference equations.
thm-ordered-monomial-pbw-basis-for-the-shifted-yangian: shifted Yangian PBW. Auditor reports partial proof only; do not treat as a completed logical supplier until local proof completion is verified.

Conventions, diagrams, and sources

Apply the definition and diagram contracts and prerequisite and repair holds before treating any proposed result as a supplier. All diagrammatic operations require defined domains, codomains, orientations, coefficients, and algebraic verification. Existing carriers keep their canonical identities and placements. Physics-only mathematical carriers require the supported same-ID resolver route specified in those contracts; links here do not establish that route.

06-other-physics-algebras

Full page contract: placement inventory (entry hopf-hecke-and-quantum-symmetry-interfaces). The original audited research inventory and its historical audit receipt remain unchanged.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

The rational spin-1/2 spectral R-matrix

Definition

Let $V = C^{2}$ with ordered basis $e_{1}, e_{2}$ . Tensor products are over $C$ and tensor powers are left-associated, as in Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. The elementary tensors $e_{a} \otimes e_{b}$ form a basis of $V \otimes V$ , and the elementary tensors $e_{a} \otimes e_{b} \otimes e_{c}$ form a basis of $V^{\otimes 3}$ , by The elementary tensors of two bases form the product basis of the tensor product.

Define the flip $P$ on $V \otimes V$ by $P (e_{a} \otimes e_{b}) = e_{b} \otimes e_{a}$ and extend $C$ -linearly. For $u \in C$ set $R (u) = u I + i P$ and $\overset{ˇ}{R} (u) = P R (u) = u P + i I$ , where $I$ denotes the identity on $V \otimes V$ .

On $V^{\otimes 3}$ define $P_{12}$ , $P_{13}$ , and $P_{23}$ on the displayed basis by swapping the indicated tensor positions and leaving the third position fixed. Put $R_{i j} (u) = u I + i P_{i j}$ . Put ${\overset{ˇ}{R}}_{1} (u) = P_{12} R_{12} (u)$ and ${\overset{ˇ}{R}}_{2} (u) = P_{23} R_{23} (u)$ . Products of endomorphisms mean composition, with the rightmost factor applied first.

This is a family of finite-dimensional linear maps. The definition asserts neither invertibility for every u nor a Hopf-algebra or categorical braiding structure.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The rational spin-1/2 spectral R-matrix satisfies Yang-Baxter

Statement

Let $V$ , $R (u)$ , $R_{i j} (u)$ , and ${\overset{ˇ}{R}}_{j} (u)$ be as in The rational spin-1/2 spectral R-matrix. For arbitrary $x, y, z \in C$ , set $u = x - y$ , $v = y - z$ , and $w = x - z = u + v$ . In ${End}_{C} (V^{\otimes 3})$ the un-checked operators satisfy

$R_{12} (u) R_{13} (w) R_{23} (v) = R_{23} (v) R_{13} (w) R_{12} (u) .$

The checked operators also satisfy the braid-form spectral identity

${\overset{ˇ}{R}}_{1} (u) {\overset{ˇ}{R}}_{2} (w) {\overset{ˇ}{R}}_{1} (v) = {\overset{ˇ}{R}}_{2} (v) {\overset{ˇ}{R}}_{1} (w) {\overset{ˇ}{R}}_{2} (u) .$

Facts & Assumptions

Given: The fixed space $V = C^{2}$ and the endomorphisms defined in The rational spin-1/2 spectral R-matrix, together with arbitrary $x, y, z \in C$ .

[F1]

The rational spin-1/2 spectral R-matrix: $P_{12}$ , $P_{13}$ , and $P_{23}$ act on the standard tensor basis by the corresponding transpositions; $R_{i j} (t) = t I + i P_{i j}$ and ${\overset{ˇ}{R}}_{1} (t) = t P_{12} + i I$ , ${\overset{ˇ}{R}}_{2} (t) = t P_{23} + i I$ .

Proof

1.1givenF1algebra

Write $s = P_{12}$ , $t = P_{13}$ , and $r = P_{23}$ . On each basis tensor $e_{a} \otimes e_{b} \otimes e_{c}$ , the products $s t$ , $t r$ , and $r s$ all give $e_{b} \otimes e_{c} \otimes e_{a}$ ; the products $s r$ , $r t$ , and $t s$ all give $e_{c} \otimes e_{a} \otimes e_{b}$ ; and $s t r$ and $r t s$ both give $e_{c} \otimes e_{b} \otimes e_{a}$ . Also $s r s$ and $r s r$ both give $e_{c} \otimes e_{b} \otimes e_{a}$ , while $s^{2} = r^{2} = I$ . Since the eight elementary tensors form a basis, these are equalities of endomorphisms.

2.1givenF1step 1.1algebra

Expand the left side of the un-checked identity as $u w v I + i (w v s + u v t + u w r) - (v s t + w s r + u t r) - i s t r$ . Expand the right side as $u w v I + i (w v s + u v t + u w r) - (u r t + w r s + v t s) - i r t s$ . By step 1.1, put $A = s t = t r = r s$ and $B = s r = r t = t s$ ; then the left quadratic term is $- (u + v) A - w B = - w (A + B)$ , and the right quadratic term is $- w A - (u + v) B = - w (A + B)$ . The scalar, linear, and cubic terms agree as well, using $u + v = w$ and $s t r = r t s$ . This proves the un-checked identity.

3.1givenF1step 1.1algebra∎

The checked left side expands to $i (u v - 1) I - w (s + r) + i w (u s r + v r s) + u v w s r s$ . The checked right side expands to $i (u v - 1) I - w (s + r) + i w (v r s + u s r) + u v w r s r$ . These expressions agree by step 1.1 and $w = u + v$ , proving the checked identity.

Diagram

Read each arrow label as a composite of endomorphisms, with the rightmost factor applied first. The two arrows have the same source and target; this is a notation diagram for the checked identity above, not part of its proof.

5 · Examples, counterexamples and false statements

None yet.