Quantum doubles, Hecke quotients and physical braid operators

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.

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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.

Let $\mathcal{C}$ be a unitary spherical fusion category over $\mathbb{C}$. Let $a,b,c$ be simple objects, with $a$ self-dual, $N_{aa}^{\mathbf{1}}=1$, and supply the normalized vacuum-channel self-duality datum with sign ${\kappa }_a\in \{+1,-1\}$ and positive spherical dimension $d_a$. For $n\ge 3$, take the fixed-boundary repeated-$a$ path space $H_{a,n;b,c}={\mathrm{Hom}}_{\mathcal{C}}(c,T_n)$ with $T_0=b$ and $T_{j+1}=T_j\otimes a$, as in Fixed-boundary anyonic fusion-path space. Let $P_i=P_{i,\mathbf{1}}$ be the orthogonal local vacuum-channel projectors and put $U_i={\delta }_a{P}_i$, where ${\delta }_a={\kappa }_a{d}_a$. Then, for every $1\le i\le n-2$,

$U_{i+1}{U}_i{U}_{i+1}=U_{i+1}\mathrm{.}$

This supplies the reverse adjacent relation for the repeated-charge Temperley–Lieb family.

Fixed-boundary anyonic fusion-path space

Local anyonic channel projector

Local channel projections are orthogonal idempotents

Frobenius–Schur sign in a normalized anyonic vacuum channel

The vacuum F-coefficient records the Frobenius–Schur sign

Canonical F-move paths give the same channel operator

The published anyonic projector proposition supplies the square relation, the left-oriented adjacent relation, and distant commutation, but its Statement does not list this reverse adjacent relation. The rank-one calculation above proves that missing direction from the same normalized F-coefficient, so the Hecke braid theorem can be applied to the actual anyonic family without assuming the omitted relation.

For a complex vector space $V$, let ${\mathrm{End}}_{\mathbb{C}}(V)$ be its set of complex-linear endomorphisms, with composition as multiplication, and define $\mathrm{GL}(V)$ to be the group of bijective elements of ${\mathrm{End}}_{\mathbb{C}}(V)$. A complex Hilbert space is a complete complex inner-product space; here the inner product is linear in its first variable and $\mathrm{\parallel }x\mathrm{\parallel }:=\sqrt{⟨x,x⟩}$. A linear operator $A:V\to V$ on such a space is bounded if some $C<\mathrm{\infty }$ satisfies $\mathrm{\parallel }Ax\mathrm{\parallel }\le C\mathrm{\parallel }x\mathrm{\parallel }$ for every $x\in V$, self-adjoint if $⟨Ax,y⟩=⟨x,Ay⟩$ for all $x,y\in V$, and unitary if it is surjective and preserves the inner product. In the formulas below, $I_V$ denotes the identity map of $V$.

Let $n\ge 2$, let $V$ be a complex vector space, and let $U_1,\dots ,U_{n-1}\in {\mathrm{End}}_{\mathbb{C}}(V)$. Suppose $\delta \in \mathbb{C}$, $q\in {\mathbb{C}}^{\times }$, and $\delta =q+q^{-1}$, and assume

$U_i^2=\delta U_i(1\le i<n),U_i{U}_{i+1}{U}_i=U_i,U_{i+1}{U}_i{U}_{i+1}=U_{i+1}(1\le i\le n-2),U_i{U}_j=U_j{U}_i(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$

Define $B_i:=q{I}_V-U_i$. Then each $B_i$ is invertible with $B_i^{-1}=q^{-1}{I}_V-U_i$; the $B_i$ satisfy the Artin braid and far-commutation relations; and

$(B_i-q{I}_V)(B_i+q^{-1}{I}_V)=0.$

Thus ${\sigma }_i\mapsto B_i$ defines a group homomorphism $B_n\to \mathrm{GL}(V)$. If $H_v(n)$ is the generic type-A Hecke algebra with $(T_i-v)(T_i+1)=0$, its scalar specialization $H_{q^2}(n)$ has a unital algebra representation $T_i\mapsto q{B}_i$. This is the standard Hecke normalization because $(q{B}_i-q^2{I}_V)(q{B}_i+I_V)=0$. If $V$ is a complex Hilbert space and each $U_i$ is bounded self-adjoint, then $\delta \in \mathbb{R}$ and $\mathrm{\mid }q\mathrm{\mid }=1$ imply that each $B_i$ is unitary. No faithfulness, diagonalizability, or physical realization is asserted.

Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations

Vacuum projectors obey the reverse adjacent Temperley–Lieb relation

The generic type-A Hecke algebra

The Hecke generators satisfy the Artin relations and are units

The braid group by Artin presentation

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

The finite-dimensional anyonic path-space operators are one algebraic source of the $U_i$. The conclusion is conditional on their displayed relations and signed loop parameter. It does not identify these operators with an experimentally realized adiabatic exchange.

Let $n\ge 2$, $V$ be a complex vector space, $U_i\in {\mathrm{End}}_{\mathbb{C}}(V)$, $\delta \in \mathbb{C}$, and $q\in {\mathbb{C}}^{\times }$ satisfy $\delta =q+q^{-1}$ and

$U_i^2=\delta U_i,U_i{U}_{i+1}{U}_i=U_i,U_{i+1}{U}_i{U}_{i+1}=U_{i+1},U_i{U}_j=U_j{U}_i(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$

Put $B_i=q{I}_V-U_i$, and let ${\rho }_n:B_n\to \mathrm{GL}(V)$ be the homomorphism ${\rho }_n({\sigma }_i)=B_i$ from the preceding Temperley–Lieb–Hecke construction. At $q=1$ one has $\delta =2$ and $B_i=I_V-U_i$. These operators satisfy $B_i^2=I_V$, so ${\rho }_n$ factors through the canonical quotient ${\pi }_n:B_n\to S_n$ as a representation $S_n\to \mathrm{GL}(V)$. The $q=1$ Hecke generator relations are then the type-A Coxeter relations. For general $q$, the roots of the quadratic relation are $q$ and $-q^{-1}$; they coincide when $q^2=-1$, and no diagonalizability is asserted. Involutivity is not imposed by the Hecke relation and need not hold.

Temperley–Lieb projectors give a Hecke braid representation

The generic type-A Hecke algebra

The symmetric group has the Coxeter presentation

The braid group surjects onto the symmetric group

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

At $q=1$ the Coxeter quotient is obtained by adding involutions to the braid presentation. At general $q$, the representation remains a braid/Hecke representation; no symmetric-group action follows unless the involution relations are separately verified.