Graph Planar Algebra

• Graph planar algebra is a diagrammatic, combinatorial *-algebra constructed from weighted bipartite graphs that encode the standard invariant of finite-index subfactors.
• It provides an explicit operator-theoretic framework via Hilbert spaces and path-concatenation, ensuring isotopy invariance and adherence to operad gluing axioms.
• The structure enables embedding of abstract planar algebras, construction of fixed-point subalgebras under automorphism groups, and classification of both classical and exotic subfactors.

A graph planar algebra is a diagrammatic, combinatorial planar *-algebra constructed from a bipartite (or, more generally, weighted and possibly oriented) graph. It is a central object in the theory of subfactor planar algebras as introduced by Jones, encoding the standard invariant of a finite-index subfactor and serving both as a universal receptacle for embeddings of abstract planar algebras and as a powerful tool for constructing and classifying subfactors and module categories. The canonical construction based on a locally finite bipartite graph with Perron–Frobenius weighting facilitates an explicit algebraic and operator-theoretic framework, underlying both deep classification results and the emergence of "exotic" structures across operator algebras, quantum groups, and low-dimensional topology (Burstein, 2010, Jones et al., 2010, Bisch et al., 2024).

1. Construction of the Bipartite Graph Planar Algebra

Let $\Gamma$ be a locally finite bipartite graph with vertex set $V=V^+\cup V^-$ and edge set $E$. One fixes a weight (spin vector) $\mu:V\to(0,\infty)$ subject to the boundedness condition: $\exists\,M<\infty$ such that $\mu(v)/\mu(w)<M$ whenever $\{v,w\}\in E$. For each $n\geq 0$ and $\pm$, define

$$P_{n,\pm} = \{\,A : \operatorname{Loops}^{\pm}_{2n}(\Gamma) \to \mathbb{C} \mid A \text{ bounded}\,\},$$

where $\operatorname{Loops}^{\pm}_{2n}(\Gamma)$ denotes closed walks of length $2n$ based at a vertex in $V^{\pm}$. The total graph planar algebra is $P(\Gamma,\mu)=\bigoplus_{n,\pm} P_{n,\pm}$, with a grading respected by the planar operad.

Given a shaded planar tangle $T$ with $k$ internal disks of types $(n_i,\epsilon_i)$ and external type $(n,\pm)$, the multilinear operad action $Z_T$ is defined via a state sum: one sums over all possible labelings $\sigma$ of regions by vertices (white by $V^+$, black by $V^-$) and contour strings by edges, requiring boundary compatibility. Each internal region (disk $D_i$) yields a closed loop $L_i$; each local maximum or minimum (singularity) contributes a ratio weight $\mu(v)/\mu(w)$ of its adjacent regions. The output is

$$Z_T(x_1,\ldots,x_k)(L) = \sum_{\sigma\,\mathrm{compatible\,with}\,L} w(\sigma), \quad w(\sigma) = \left(\prod_\mathrm{singularities\,s} \frac{\mu(v_s)}{\mu(w_s)}\right) \prod_{i=1}^k x_i(L_i).$$

One checks that only finitely many states contribute and that the resulting maps satisfy isotopy invariance, *-structure, and the operad gluing axioms, making $P(\Gamma,\mu)$ a planar *-algebra (Burstein, 2010, Jones et al., 2010).

The modulus $\delta$ of $(\Gamma,\mu)$ is defined by demanding shaded and unshaded circles evaluate identically: $$\sum_{e:s(e)=v} \mu(t(e))^2 = \delta\,\mu(v)^2,\quad \forall\,v.$$ For finite $\Gamma$ this is the Perron–Frobenius eigenvalue of the adjacency matrix.

2. Algebraic and Operator-Theoretic Realizations

The $n$-box spaces $P_{n,\pm}$ realize as spaces of bounded operators on Hilbert spaces $H_n$ of square-summable length-$n$ paths starting at vertices of fixed parity. The entire planar algebra acts as a concrete collection of operators commuting with left and right projections corresponding to source and target vertices. The operad multiplication, inclusion, capping, and involution operations are exactly implemented by path-concatenation, contraction, and reversal on $H = \bigoplus H_n$ (Burstein, 2010).

Graph planar algebras are $C^*$-algebras when $\Gamma$ is finite and $\mu$ satisfies the eigenvector condition, yielding finite-dimensional semisimple *-algebras, positive definite traces, and spherical structure. In the oriented (or weighted) setting, the construction extends by labeling regions from suitable branching graphs and weighting the state sum by generalized harmonic functions (Koshida, 2023).

3. Automorphism Groups and Fixed-Point Subalgebras

The automorphism group of a bipartite graph planar algebra $P(\Gamma,\mu)$ is classified as follows:

• Graph-automorphism operators: Any automorphism $\kappa$ of $\Gamma$ (preserving $V^+$, $V^-$, adjacency, and scaling $\mu$ by a constant) yields a unitary $U_\kappa$ acting by permuting path-basis vectors.
• Multiplication operators: A unitary $O\in B(H_1)$ commuting with source/target projections and satisfying $\mathrm{rev}\, O\, \mathrm{rev} = O^*$ may be extended to a grading-preserving unitary $U_O$ commuting with concatenation and reversal, acting on loops by twisting by $1$-dimensional representations of $\pi_1(\Gamma)$.

The group of all planar *-automorphisms $\operatorname{Aut}(P)$ factors uniquely as $E\rtimes A$, where $E$ is the group of multiplication operators (isomorphic to the dual of $H^1(\Gamma)$), and $A$ is the group of automorphisms of $\Gamma$ (Burstein, 2010).

For any subgroup $G\leq\operatorname{Aut}(P)$, the fixed points $$P^G_{n,\pm} = \{x\in P_{n,\pm}\mid g(x)=x \ \forall\, g\in G\}$$ form a planar *-subalgebra. If $G$ acts transitively on $V^+$ and $V^-$, $P^G$ is a subfactor planar algebra with the same modulus as $P$, realized as the standard invariant of a finite-index extremal II$_1$ subfactor. The index of the corresponding subfactor is $\delta^2$ (Burstein, 2010).

4. Structural Theorems and Embedding Results

The bipartite graph planar algebra $P(\Gamma,\mu)$ arises as the relative commutant planar algebra for the basic construction tower of a strongly Markov inclusion of finite-dimensional algebras with Bratteli diagram $\Gamma$. There is a canonical *-isomorphism between the tower's relative commutant planar algebra and $P(\Gamma,\mu)$ (Jones et al., 2010).

Every finite-depth subfactor planar algebra $Q_{\bullet}$ of modulus $d$ embeds as a planar *-subalgebra of the graph planar algebra $P(\Gamma)$ of its principal graph $\Gamma$. This embedding identifies the Temperley–Lieb generators and respects all operad tangles (Jones et al., 2010, Morrison, 2013, Bisch et al., 2024).

The same mechanism, extended to commuting square subfactors and more general weighted or oriented graphs (e.g., Young graphs with harmonic functions), yields powerful obstructions for standard invariants and organizes the construction of subfactors with prescribed combinatorial and analytic properties (Bisch et al., 2024, Koshida, 2023).

5. Applications and Classification Frameworks

The fixed-point planar subalgebra construction under automorphism subgroups provides uniform realizations for a range of classical and exotic subfactors:

• Group-subgroup (star) subfactors: $\Gamma$ is a star graph, $G\leq S_n$ acts on leaves; $P^G$ recovers the standard invariant of $M^G\subset M$, index $n$.
• Diagonal subfactors: Cayley-type bipartite graphs encode automorphism-generated inclusions.
• Bisch–Haagerup and Wassermann subfactors: $\Gamma$ reflects group-theoretic and module-coset structures, with $G$ acting by graph or multiplication operators (Burstein, 2010, Brothier, 2016).
• Infinite-depth and non-amenable cases: Tree graphs and suitable automorphism actions yield subfactor planar algebras with Haagerup property and complete metric approximation property (CMAP) (Brothier, 2015).

In the broader context, every symmetric enveloping inclusion of a subfactor planar algebra arising from a Hecke pair $(G,K)$ (where $K$ is a compact open almost normal subgroup of $G$) can be described as a crossed product, providing analytic control over amenability, approximation, and representation-theoretic properties (Brothier, 2016, Brothier, 2015).

6. Computational and Category-Theoretic Connections

Graph planar algebra embeddings are used to facilitate explicit computation and the construction of new diagrammatic relations in the study of module categories over quantum groups and fusion categories. By embedding the planar algebra of a generating object into the graph planar algebra of its fusion graph, one translates skein-theoretic constraints into concrete finite-dimensional linear algebra, enabling the discovery of new quantum subgroups, such as the type $G_2$ quantum subgroups via computational solution of relations among diagrams in the ambient graph planar algebra (Hill, 8 Jan 2026).

Moreover, the construction extends via oriented branching graphs, harmonic functions, and combinatorial data to categorification contexts such as the Khovanov Heisenberg category, with explicit state sums on tangles recovering the relevant algebraic and representation-theory invariants (Koshida, 2023).

7. Impact and Open Directions

The graph planar algebra acts as a universal combinatorial host for subfactor planar algebras, unifying their construction, providing embedding theorems, and enabling new analytic and combinatorial approaches to classification. Key impacts include:

• Transparent derivation of known and "exotic" subfactors via group action fixed points.
• Uniform strategy for enforcing finiteness and sphericality via symmetry.
• Concrete analytic inheritance of approximation properties (amenability, Haagerup, CMAP) from acting groups to fixed-point subfactors (Brothier, 2015).
• Obstructions for principal graphs of subfactors via embedding criteria, guiding the classification of standard invariants (Morrison, 2013).

Open problems include the detailed classification of which automorphism group actions lead to non-isomorphic fixed-point planar subalgebras, the construction of amenable or non-amenable subfactors from infinite graphs, and the systematic generation of continuous families of examples via $\pi_1(\Gamma)$ character twisting. Further computational developments—leveraging the diagrammatic presentation and the translation to module category theory—continue to expand the reach of graph planar algebra techniques (Burstein, 2010, Hill, 8 Jan 2026).