Principal Weights in Planar Algebras

Updated 30 June 2025

Principal weights are specialized weighting structures that assign positive scalars to components in planar algebras and bimodules, capturing core algebraic properties.
They utilize weight functions on principal graphs to encode fusion rules by ensuring that weights multiply consistently along tensor products.
Their constructive perturbation methods preserve key invariants like sphericality and fusion stability, offering robust tools for subfactor theory research.

Principal weights refer to specifically defined weighting structures that play a central role across several areas of mathematics, coding theory, machine learning, and representation theory. In contemporary research, the term typically arises in contexts where weights are assigned in a manner reflecting principal objects (such as principal components, principal ideals, or principal representations), thereby informing key algebraic, analytic, or algorithmic behavior. The following sections synthesize major advances and operational characterizations of principal weights from foundational research on bimodules and planar algebras (Das et al., 2010 ).

1. Weights in Planar Algebras and Principal Weights

The concept of a weight on a planar algebra, introduced in subfactor theory, involves an invertible element $z \in P_{+1}$ of the degree-one positive part of the planar algebra $P$ such that for every color $k$ (labeled as $(\pm, k)$ ), the induced elements $z_{+k}$ and $z_{-k}$ are central in their respective degree- $k$ parts $P_k$ . These elements are defined diagrammatically by the planar algebra's action, with centrality ensuring that the alteration imposed by the weight commutes with each component of the algebra. The assignment of such weights can be viewed as "twisting" the algebraic structure, with consequential changes in invariants like isotopy or trace.

2. Weight Functions and Principal Graphs

The main theoretical development is the characterization of weights via weight functions on the vertices of the principal graphs associated with a bimodule or a subfactor. The principal graph encodes the inclusion and fusion relations among irreducible bimodules at each tensor power level. A principal weight in this context corresponds to a function $w : V_{-, \eta} \to (0,\infty)$ assigning a positive number to every isomorphism class of irreducible $(-, \eta)$ -bimodules (vertices of the principal graph).

The critical property satisfied by principal weight functions is the tensor homomorphism property: $w_{v_3} = w_{v_1} w_{v_2}$ whenever $v_3$ is an irreducible summand of the tensor product $v_1 \otimes v_2$ . This ensures that principal weights respect the categorical tensor structure, encoding how weights multiply along fusion paths in the associated tensor category. The research establishes a one-to-one correspondence between positive weights on the planar algebra and weight functions on the entire bicategory of bimodules generated by a given bimodule.

3. Trivial Perturbation Class (TPC) and Fusion Stability

A central property in the theory of bimodules is the notion of a trivial perturbation class (TPC). A planar algebra $P$ or the corresponding bimodule is said to have TPC if all positive weight perturbations yield spherical algebras (i.e., those for which the normalized left and right traces on $P_{+1}$ coincide, preserving isotopy invariance).

The research demonstrates that TPC is closed under Connes fusion, a fundamental operation corresponding to the relative tensor product (fusion) of bimodules. Explicitly, if two bifinite bimodules each enjoy TPC, so does their Connes fusion. This closure is established by propagating the tensor homomorphism property: the only compatible weight functions for the fused bimodule when each constituent is trivial are those that assign the constant value 1, thereby preserving sphericity.

4. Direct Construction of Perturbed Bimodules

The paper introduces a constructive method to perturb a bifinite bimodule by a prescribed positive weight such that the associated bimodule planar algebra is isomorphic to the perturbation (by the same weight) of the original planar algebra. The construction proceeds as follows:

Given a bifinite bimodule $_{A_+} H_{A_-}$ with associated planar algebra $P$ and a positive weight $z$ , determine the induced weight function $w$ on the principal graphs.
For each minimal central projection $s$ in $P_{+1}$ , associate an $R$ - $R$ -bimodule $K^{w_s}$ with dimension $w_s$ (using a hyperfinite II $_1$ factor $R$ ).
Form the perturbed bimodule: $H' = \bigoplus_{s \in S} K^{w_s} \otimes \mathrm{Range}~s$ with new acting algebras $A'_+ = R \otimes A_+$ and $A'_- = R \otimes A_-$ .

The main result is that the planar algebra constructed from $_{A'_+} H'_{A'_-}$ is isomorphic to the perturbation of $P$ by $z$ , realized by explicit intertwiner and isometry construction for each graded component.

5. Mathematical Formulas and Operational Summary

Key structures and relationships formulated in the research include:

Weight function property: $w_{v_3} = w_{v_1} w_{v_2}$ for fusion subobjects.
Perturbed bimodule construction: $H' = \bigoplus_{s \in S} K^{w_s} \otimes \mathrm{Range}~s$ , where $S$ indexes the minimal central projections.
TPC criterion: $P$ has TPC precisely when all weight functions are trivial: $w \equiv 1$ .

6. Structural and Theoretical Impact

The explicit characterization of principal weights via combinatorial data on the principal graphs provides computable and verifiable criteria for the existence and uniqueness of weight structures on planar algebras and their bimodules. The preservation of TPC under Connes fusion ensures that critical properties, such as sphericity, are robust under composition, with direct consequences for the construction of higher depth or composite subfactor bimodules.

The constructive approach to perturbing bimodules by principal weights gives practitioners a concrete mechanism for tuning the properties of bimodule categories, with guarantees on algebraic isomorphism and preservation of key invariants such as hyperfiniteness. These results also reveal that in many important families, including certain infinite depth or reducible planar algebras, the only allowable principal weights are scalars, ensuring the universal presence of TPC.

Summary Table: Key Structures in Principal Weights for Bimodule Planar Algebras

Object	Defining Feature	Key Formula / Property
Weight on planar algebra	Central invertible $z \in P_{+1}$ for all colors	Centrality in each graded part
Weight function on graph	$w_v \in (0, \infty)$ for each principal graph vertex	$w_{v_3}=w_{v_1} w_{v_2}$ for fusion : $v_3 \leq v_1\otimes v_2$
TPC	All positive weight perturbations are spherical	All weight functions trivial ( $w \equiv 1$ )
Perturbed bimodule	Direct sum using weights for each summand	$H' = \bigoplus_{s} K^{w_s} \otimes \mathrm{Range}~s$
Fusion stability	Closedness of TPC under Connes fusion	Tensor homomorphism forces constant weight in fusion

Principal weights thus serve as central invariants governing perturbations, isotopy classes, and categorical fusion in the theory of bimodules and planar algebras. Their explicit definition via principal graphs and categorical structures links graphical and algebraic approaches, offering robust tools for constructing and analyzing subfactor-related algebraic systems.

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