Braided line representation data

Let $k$ be a field and let $\mathcal{C}$ be a locally finite $k$-linear abelian rigid monoidal category whose tensor product is $k$-bilinear. For each object $Y$ write ${\mathrm{End}}_{\mathcal{C}}(Y):={\mathrm{Hom}}_{\mathcal{C}}(Y,Y)$. The tensor unit $\mathbf{1}$ is assumed to satisfy ${\mathrm{End}}_{\mathcal{C}}(\mathbf{1})\cong k$ as $k$-algebras. Here rigid means every object has left and right duals with evaluation and coevaluation maps satisfying the two snake identities. Equip $\mathcal{C}$ with a braiding $c_{Y,Z}:Y\otimes Z\to Z\otimes Y$, natural in $Y,Z$, satisfying the two hexagon identities with the associator $\alpha$. Fix $X\in \mathcal{C}$, put $P_0(X)=\mathbf{1}$, $P_1(X)=X$, and recursively put $P_{m+1}(X)=P_m(X)\otimes X$.

For any object $Y$, let ${\mathrm{Aut}}_{\mathcal{C}}(Y)$ be the group under composition of those $g\in {\mathrm{End}}_{\mathcal{C}}(Y)$ for which some $h\in {\mathrm{End}}_{\mathcal{C}}(Y)$ satisfies $g\circ h=h\circ g={\mathrm{id}}_Y$.

For each $n\ge 2$ and $1\le i<n$, let $Q_{n,i}(X)$ be the parenthesized product $(P_{i-1}(X)\otimes (X\otimes X))\otimes P_{n-i-1}(X)$, with the unit factors at either end removed by the fixed unitors when an exponent is zero. Choose a canonical rebracketing isomorphism ${\rho }_{n,i}:P_n(X)\to Q_{n,i}(X)$ made from associators, unitors, and their inverses, without permuting the ordered tensor factors. Define

$B_i^X:={\rho }_{n,i}^{-1}\circ ({\mathrm{id}}_{P_{i-1}(X)}\otimes c_{X,X}\otimes {\mathrm{id}}_{P_{n-i-1}(X)})\circ {\rho }_{n,i}\in {\mathrm{End}}_{\mathcal{C}}(P_n(X))\mathrm{.}$

The family $B_i^X$ is independent of the chosen canonical rebracketing path and determines a homomorphism ${\varrho }_n:B_n\to {\mathrm{Aut}}_{\mathcal{C}}(P_n(X))$, where $B_n$ is the braid group with positive generator ${\sigma }_i$ sent to $B_i^X$. These are exact algebraic morphisms.

Physical interpretation boundary. A physical adiabatic-exchange interpretation needs additional model data; the categorical data below supply none of the analytic or physical data. Here $\mathcal{H}$ is a complete complex inner-product space, with inner product linear in its first variable. For each dimensionless $s\in [0,1]$, let $K(s):\mathcal{D}\subseteq \mathcal{H}\to \mathcal{H}$ be self-adjoint on a specified common dense domain $\mathcal{D}$. For a densely defined operator $K$, its adjoint $K^{\ast }$ is defined by $y\in \mathcal{D}(K^{\ast })$ exactly when some $z\in \mathcal{H}$ satisfies $⟨Kx,y⟩=⟨x,z⟩$ for every $x\in \mathcal{D}(K)$, with $K^{\ast }y=z$; self-adjoint means $\mathcal{D}(K^{\ast })=\mathcal{D}$ and $K^{\ast }x=Kx$ there. The spectrum $\sigma (K)$ is the set of $\lambda \in \mathbb{C}$ for which $K-\lambda I$ has no bounded, everywhere-defined inverse.

For each $s$, supply a projection-valued measure ${\mathsf{E}}_s$ on the Borel subsets of $\mathbb{R}$: each ${\mathsf{E}}_s(A)$ is an orthogonal projection, ${\mathsf{E}}_s(\mathbb{R})=I$, and for pairwise disjoint Borel sets $A_j$, ${\mathsf{E}}_s({\bigcup }_j{A}_j)x={\sum }_j{\mathsf{E}}_s(A_j)x$ with norm convergence for every $x\in \mathcal{H}$. Require ${\mathsf{E}}_s$ to represent $K(s)$ as its spectral integral: with ${\mu }_{s,x}(A):=⟨{\mathsf{E}}_s(A)x,x⟩$, the domain is $\mathcal{D}=\{x\in \mathcal{H}:{\int }_{\mathbb{R}}{\lambda }^{2}d{\mu }_{s,x}(\lambda )<\mathrm{\infty }\}$ and $K(s)x={\int }_{\mathbb{R}}\lambda d{\mathsf{E}}_s(\lambda )x$ there. For a Borel set $\mathrm{\Omega }\subseteq \mathbb{R}$, define its spectral projection by $P(s):={\mathsf{E}}_s(\mathrm{\Omega })$. Require $\mathrm{ran}P(s)$ to be finite-dimensional and $\mathrm{dist}(\mathrm{\Omega },\sigma (K(s))\setminus \mathrm{\Omega })\ge \gamma >0$ uniformly in $s$, where $\mathrm{dist}(A,B):=\inf \{\mathrm{\mid }a-b\mathrm{\mid }:a\in A,b\in B\}$ and $\inf \mathrm{\varnothing }:=+\mathrm{\infty }$.

Require $s\mapsto P(s)$ to be continuously differentiable in operator norm. Its ranges then form a finite-rank Hermitian vector bundle $E\to [0,1]$, meaning the locally trivial bundle with fiber $E_s:=\mathrm{ran}P(s)$ and the inner product inherited from $\mathcal{H}$. A Hermitian connection on $E$ is a complex-linear covariant derivative ${\mathrm{\nabla }}_s$ on continuously differentiable sections satisfying ${\mathrm{\nabla }}_s(f\psi )=f^{\mathrm{\prime }}\psi +f{\mathrm{\nabla }}_s\psi$ and ${\mathrm{\partial }}_s⟨\psi ,\varphi ⟩=⟨{\mathrm{\nabla }}_s\psi ,\varphi ⟩+⟨\psi ,{\mathrm{\nabla }}_s\varphi ⟩$. Its parallel transport $W(s,0):E_0\to E_s$ is the map obtained by solving ${\mathrm{\nabla }}_s\psi (s)=0$ along the interval.

Supply a family of unitary operators $U_{\epsilon }(s,0):\mathcal{H}\to \mathcal{H}$, where unitary means surjective and inner-product-preserving, with $U_{\epsilon }(0,0)=I$ and $U_{\epsilon }(s,0)\mathcal{D}\subseteq \mathcal{D}$. Require for each $x\in \mathcal{D}$ that $s\mapsto U_{\epsilon }(s,0)x$ is differentiable in Hilbert norm and satisfies $i\epsilon {\mathrm{\partial }}_s{U}_{\epsilon }(s,0)x=K(s)U_{\epsilon }(s,0)x$; all conditions ensuring this propagator exists are part of the model data. Here $\epsilon >0$ is an energy scale and the slow-evolution regime means $\epsilon \downarrow 0$. A linear operator $A$ on $\mathcal{H}$ is bounded when ${\sup }_{\mathrm{\parallel }x\mathrm{\parallel }\le 1}\mathrm{\parallel }Ax\mathrm{\parallel }<\mathrm{\infty }$; this supremum is its operator norm $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. The added data must include a bound $r(\epsilon )\to 0$ such that

${\sup }_{s\in [0,1]}\mathrm{\parallel }{U}_{\epsilon }(s,0)P(0)-{\iota }_{s}W(s,0)P(0){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le r(\epsilon ),$

where ${\iota }_s:E_s\hookrightarrow \mathcal{H}$ is inclusion. In a physical realization, $K(s)$, $\gamma$, and $\epsilon$ have the model's energy units; no numerical values are fixed here. The categorical data do not assert an experimental realization.

Tensor and multitensor categories

Braiding

Braided coherence is controlled by underlying braids

The braid group by Artin presentation

Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group

Finite double and its canonical quasitriangular structure

The first diagram is a composition diagram: arrows run from source to target, and the outer path is the definition $B_i^X={\rho }_{n,i}^{-1}{C}_i{\rho }_{n,i}$, where $C_i$ is the local tensor of $c_{X,X}$ with spectator identities. The label ${\rho }_{n,i}(\alpha )$ means a finite composite of associators and unitors. In the second diagram time/order points upward; the positive crossing is, by definition, $c_{X,X}$, with the right input strand passing over the left. Both drawings omit spectator strands. They record conventions only; the proof uses morphism composition and coherence, not the diagrams.

diagram failed to render:
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  \draw (1.4,0) .. controls (1.4,1.1) and (0,1.9) .. (0,3);
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  \node at (0, 3.3) {$X$}; \node at (1.4, 3.3) {$X$};
  \node at (0.7,1.5) {$c_{X,X}$};
  \draw[->] (2.1,0.2) -- (2.1,2.8) node[midway,right] {$\text{time}$};
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The finite-double result is used only at its inspected interface. The definition does not assert that an arbitrary quasitriangular Hopf algebra is semisimple or unitary, that it defines a modular tensor category, or that its modules are realized by an experiment.