Semisimple, Nonsemisimple, and Unitarity Boundaries

Semisimplicity, rigidity, pivotality, braiding, and unitarity are independent hypotheses. Fusion-category theorems use finite semisimplicity to split every object into simples; unitary modular-category formulas additionally use a positive dagger structure and nondegenerate braiding. In nonsemisimple finite tensor categories, indecomposable extensions and projective objects survive, ordinary categorical traces may vanish, and modified traces on tensor ideals replace some—but not all—semisimple dimension formulas.

Required background. DHR Sectors and Modular Tensor Categories of Nets supplies a unitary semisimple benchmark. Fusion Categories, Module Categories, and Bimodule Defects states the fusion hypotheses. Noninvertible Symmetries, Fusion, and Junction Data supplies quantum dimensions and $F$-symbols. Helpful background. Logarithmic CFT and Indecomposable Modules gives the physical nonsemisimple setting, while Nonunitary CFTs, Effective Central Charge, and Complex Data separates nonunitarity from logarithmic structure.

The hypothesis boundary

A fusion category is finite, semisimple, and rigid with simple unit. A finite tensor category retains finite length, rigidity, and exact tensor product but may be nonsemisimple. Pivotality identifies the double-dual functor with the identity coherently. Sphericality equates left and right traces. A braiding supplies crossings; modularity requires a nondegeneracy condition. Unitarity supplies a positive dagger compatible with tensor product and makes the structural isomorphisms unitary.

No implication runs automatically from rigidity to semisimplicity or from sphericality to positivity. In a semisimple unitary fusion category, every object decomposes as

$$X\cong\bigoplus_i n_iX_i,$$

and the positive quantum dimensions obey $d_id_j=\sum_kN_{ij}^{\ k}d_k$. This supports diagonal fusion analysis. In a nonsemisimple category, the Grothendieck group records composition factors but forgets extension sequences such as

$$0\longrightarrow X\longrightarrow E\longrightarrow Y\longrightarrow0$$

that do not split. A fusion matrix on simple classes cannot distinguish $E$ from $X\oplus Y$.

Ising versus a finite nonsemisimple category

The Ising category is semisimple and unitary. Its simple dimensions are $1,1,\sqrt2$, its $F$-matrices can be chosen unitary, and every object is a direct sum of $\mathbf1,\psi,\sigma$. There are no nontrivial extensions among these simples inside the fusion category.

For a small contrast, take the Sweedler Hopf algebra $H_4$ over $\mathbb C$, generated by $g,x$ with

$$g^2=1, \qquad x^2=0, \qquad gx=-xg,$$

and coproduct $\Delta g=g\otimes g$, $\Delta x=1\otimes x+x\otimes g$. Its finite-dimensional representation category is rigid and finite but not semisimple. It has two one-dimensional simples and nontrivial two-dimensional indecomposable projective covers $P_+$ and $P_-$. Their radical filtrations cannot be recovered from the Grothendieck ring.

With a pivotal structure, the ordinary categorical dimension of projective objects can vanish even when the object is nonzero. Closing a projective strand with the ordinary trace then erases information. A modified trace is a family

$$t_P:\operatorname{End}(P)\longrightarrow\mathbb C$$

on the tensor ideal of projectives, cyclic under composition and compatible with partial traces. The modified dimension is $d_t(P)=t_P(\operatorname{id}_P)$. Geer, Kujawa, and Patureau-Mirand construct traces on ideals and define modified dimensions in Geer, Kujawa, and Patureau-Mirand 2011, §§3–4, printed pp. 12–20 (PDF). Their construction supplies a replacement invariant under named hypotheses; it does not turn the category semisimple or make its dimensions positive.

The exact first application returns to Logarithmic CFT and Indecomposable Modules. Projective modules and Jordan blocks there produce logarithmic correlators. The categorical comparison explains why ordinary fusion eigenvalues miss extensions; the conformal page owns the analytic correlators and Virasoro representation theory.

Invalid exports from the semisimple case

Diagonalizing a Grothendieck fusion matrix in $\operatorname{Rep}(H_4)$ does not split the projective covers. Positive quantum-dimension entropy formulas also fail when the chosen categorical trace vanishes or changes sign. Modified dimensions may restore useful link invariants, but they depend on a trace normalization and tensor ideal and need not be positive.

The adversarial test replaces every indecomposable by the sum of its composition factors. The Grothendieck class is unchanged, so any computation using only $K_0$ passes. Yet nilpotent endomorphisms, nontrivial extensions, and Jordan blocks disappear. The strongest surviving claim is an equality in the Grothendieck group. It is not an isomorphism of objects, a spectral decomposition, or a unitary sector decomposition.

One should also distinguish nonsemisimplicity from nonunitarity. A semisimple category may lack a positive dagger, while a nonsemisimple category cannot be a unitary fusion category because orthogonal complements would split its finite-dimensional representations. Modified traces address the loss of ordinary trace information; they do not furnish a positive inner product. Positivity, semisimplicity, and nondegenerate braiding must each be checked by its own criterion.

Exercises

Why can a modified trace be nonzero when the ordinary categorical trace vanishes on projectives?

Solution

The ordinary trace is defined on the whole pivotal category and may cancel on a projective because of its nonsemisimple structure. A modified trace is instead a compatible family on the projective tensor ideal, normalized independently and constrained by cyclicity and partial-trace identities. Those conditions do not force it to equal the vanishing ordinary trace.

References

• Geer, Nathan, Jonathan Kujawa, and Bertrand Patureau-Mirand. “Generalized Trace and Modified Dimension Functions on Ribbon Categories.” Selecta Mathematica 17 (2011): 453–504. DOI; Open PDF.
• Ostrik, Victor. “Module Categories, Weak Hopf Algebras and Modular Invariants.” Transformation Groups 8 (2003): 177–206. DOI; Open PDF.