Semisimplification

Let $\mathcal C$ be a spherical $k$-linear tensor category. A morphism $f\colon X\to Y$ is negligible if

\[\label{eq:negligible-morphism} \operatorname{tr}(g\circ f)=0 \qquad\text{for every }g\colon Y\to X.\]

The negligible morphisms form a tensor ideal $\mathcal N$. The semisimplification has the same objects as $\mathcal C$ and quotient Hom spaces

\[\label{eq:semisimplification-hom} \operatorname{Hom}_{\overline{\mathcal C}}(X,Y) =\operatorname{Hom}_{\mathcal C}(X,Y)/\mathcal N(X,Y).\]

Under the hypotheses of Etingof et al. (2015), Exercise 8.18.9 and Etingof and Ostrik (2022), Proposition 2.4 and Theorem 2.6, this quotient is a semisimple tensor category. In particular, these hypotheses hold for a locally finite spherical multitensor category. The constructor in TensorCategories.jl currently accepts this case over an exact coefficient field. It also accepts a TensorPowerCategory whose ambient category satisfies these conditions.

An object $X$ becomes zero precisely when $\operatorname{id}_X$ is negligible. This condition is determined by the complete trace pairing

\[\label{eq:semisimplification-trace-pairing} \operatorname{Hom}_{\mathcal C}(X,Y)\times \operatorname{Hom}_{\mathcal C}(Y,X)\longrightarrow k, \qquad (f,g)\longmapsto\operatorname{tr}(g\circ f).\]

Consequently, $\dim(X)=0$ alone does not imply that $X$ vanishes. This distinction is essential over nonsplitting fields, where a simple object can have a nontrivial division algebra of endomorphisms. The implementation takes the radical of the full pairing in equation \eqref{eq:semisimplification-trace-pairing}. Over an algebraically closed field, the simple objects of $\overline{\mathcal C}$ are precisely the images of the indecomposable objects of nonzero categorical dimension. Over a general field, the full trace pairing is the appropriate criterion; the surviving simple objects can have division algebras of endomorphisms larger than $k$.

The interface

Two SemisimplifiedMorphism values are equal when their representatives differ by a negligible morphism. Kernels, cokernels, inverses, and coordinates are therefore computed in the quotient Hom spaces rather than from the matrices of the representatives alone.

The quotient inherits the tensor product, duality, pivotal structure, braiding, and spherical structure of the input. It is locally finite and semisimple, but it need not have only finitely many simple objects. Thus semisimplification does not by itself produce a fusion category. If the input is already fusion, the quotient reports is_fusion(Q) == true. More generally, the fusion and multifusion predicates become available once the implementation has established a finite complete list of quotient simples and checked that their endomorphism algebras are the coefficient field.

If $\mathcal C$ has finite representation type, then $\overline{\mathcal C}$ has only finitely many simple objects. It is therefore weak fusion when its unit is simple, and it is fusion when it is moreover split. The distinction disappears over an algebraically closed field. For $\mathcal C=\operatorname{Rep}_k(G)$ in characteristic $p>0$, finite representation type is equivalent to the Sylow $p$-subgroups of $G$ being cyclic (Higman, 1954).

The functions is_negligible, trace_pairing, and quotient_hom_dimension, as well as the construction of quotient Hom spaces, currently require exact coefficients and finite Hom spaces with computable bases. Enumeration of all simple objects additionally requires an implementation of indecomposables(C) for the input category; working with specified objects does not require such an enumeration.

Example: the category $\operatorname{Ver}_5$

Let $J_n$ denote the $n$-dimensional indecomposable representation of $C_5$ in characteristic $5$, on which a generator acts by one unipotent Jordan block. The projective object $J_5$ is negligible, whereas $J_1,\ldots,J_4$ survive.

using TensorCategories, Oscar
F = GF(5)
C = rep(F, cyclic_group(5))

function jordan_block(C, n)
    A = identity_matrix(base_ring(C), n)
    for i in 1:n-1
        A[i, i+1] = 1
    end
    Representation(C, gens(base_group(C)), [A])
end

J = [jordan_block(C, n) for n in 1:5]
Q = semisimplification(C)
(is_negligible(J[2]), is_negligible(J[5]), quotient_hom_dimension(J[2]))

(false, true, 1)

The surviving objects $L_n=[J_n]$ satisfy the truncated $\mathfrak{sl}_2$ tensor-product rule (Etingof et al., 2017; §2.1). For example, tensoring by $L_2$ has the adjacency matrix of the Dynkin diagram $A_4$:

L = [semisimplify(J[n], Q) for n in 1:4]
[int_dim(Hom(L[2] ⊗ L[j], L[k])) for j in 1:4, k in 1:4]

4×4 Matrix{Int64}:
 0  1  0  0
 1  0  1  0
 0  1  0  1
 0  0  1  0

This quotient is $\operatorname{Ver}_5$ (Etingof and Ostrik, 2022; Example 2.7). The dedicated constructor and bounded tensor-sector tools are described under symmetric Verlinde categories in characteristic $p$.

Continue with algebra objects and internal modules.