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César Galindo

Publications and source records attributed to César Galindo.

At least 19 recordsLinked to original sources
arXiv2026

Quotients of graded quasi-Hopf algebras

We classify the quasi-Hopf ideals of quasi-Hopf algebras faithfully graded by arbitrary groups. Each ideal determines a normal subgroup, recording which degrees are identified in the quotient, together with an invariant ideal in the corresponding neutral component. We refine this description using an ideal of the original neutral component and a retraction of a naturally associated intermediate quotient. We prove that this intermediate quotient splits as the tensor product of its neutral component with the group algebra of the normal subgroup. For fixed subgroup and neutral-component data, the possible ideals, whenever they exist, form a torsor described by equivariant homomorphisms into central group-like elements. Our results require neither finite dimensionality nor bijectivity of the antipode. We apply them to cocentral abelian extensions with possibly infinite grading, finite-dimensional neutral component, and nontrivial reassociator.

math.RA↗
arXiv2026

On Split Forms of Fusion Categories

We formulate split Galois descent for fusion categories through framed equivalences, reducing the existence of a split form to a group-theoretic splitting problem. The associated degree-two obstruction combines categorical coherence with stable descent of the simple objects, while fusion spaces impose index and parity restrictions. For quantum groups at roots of unity, we use the quasi-$R$-matrix to make the bar involution tensor-compatible and prove that the underlying fusion category $\mathcal C(\mathfrak g,k)$, associated with the simply connected root datum, has a split form over the maximal totally real subfield of the cyclotomic field generated by the quantum parameter. The same holds for all its cyclotomic Galois conjugates; in particular, every $\mathcal C(\mathfrak g,k)$ has a split real form. By contrast, no split real form exists for the standard braiding except on $\mathcal C(E_8,1)$ and $\mathcal C(D_{4m},1)$, $m \geq 1$. We further show that associative zesting can destroy real descent of $\mathcal C(\mathfrak g,k)$, while every braided zesting of $\mathcal C(\mathfrak g,k)$ still admits a split real form as an underlying fusion category. Applications to pointed and finite-group representation categories give criteria for real descent in terms of cohomology and Frobenius--Schur indicators.

math.QA↗
arXiv2026

Splitting of Clifford groups associated to finite abelian groups

The Clifford group associated with a finite abelian group fits into an extension of the symplectic group by the underlying phase space. We prove that this extension splits as a semidirect product if and only if the order of the underlying abelian group is not divisible by four. The obstruction to splitting is controlled entirely by the 2-primary component and vanishes precisely when this component is trivial or cyclic of order two. This confirms a conjecture of Korbelář and Tolar, extending their cyclic result to arbitrary finite abelian groups.

math.GR↗
arXiv2026

Representation theory of projective Clifford groups via isocategoricality

We prove that the projective Clifford group $C(A)$ associated with a finite abelian group $A$ is isocategorical to the affine symplectic group $\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$, where $V_A=A\oplus\widehat{A}$. This explicit isomorphism of tensor categories gives common parametrizations of their irreducible characters and conjugacy classes, under which their character tables agree entry by entry. It also gives an explicit basis for every tensor-power commutant of the projective Weil representation, indexed by symplectic orbits of tuples in $V_A$ whose entries sum to zero. For cyclic $A$, we classify these orbits by oriented two-generated submodules and obtain exact commutant dimensions in every tensor power.

math.RT↗
arXiv2026

Cocentral Split Abelian Hopf Algebra Extensions from Crossed Cocycles

We study cocentral algebra-split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero for a fixed action of a group on a finite abelian group. We describe the extension classes in terms of crossed families of normalized 2-cocycles and construct the obstruction to representing crossed families of cohomology classes by cocycles satisfying the crossed identity. A bicharacter obstruction maps to the strict lifting obstruction and may remain nonzero even when its image vanishes. For permutation modules, we obtain an explicit description of the corresponding cocentral extension groups. We also study finite reductions of geometric representations of Coxeter groups and compute the first cohomology of the associated linear coefficient modules for finite dihedral groups and the infinite dihedral group.

math.QA↗
arXiv2026

Fiber Functors of Equivariantizations of Finite Tensor Categories

Let $G$ be a finite group acting on a finite tensor category $\mathcal{C}$. We classify fiber functors on the equivariantization $\mathcal{C}^G$ in terms of equivariant exact module categories over $\mathcal{C}$, indexed by subgroups of $G$. The data are a subgroup $H\subseteq G$ and an $H$-equivariant $\mathcal{C}$-module category $\mathcal{M}$ whose underlying $\mathcal{C}$-module category is indecomposable, exact, and semisimple; they give a fiber functor precisely when $H$ acts transitively on the simple objects of $\mathcal{M}$ and the stabilizer cocycle of one, hence every, simple object is non-degenerate. Through Tannaka-Krein reconstruction this describes realizations of $\mathcal{C}^G$ as the representation category of a finite-dimensional Hopf algebra, with no semisimplicity hypothesis on $\mathcal{C}$. As applications, for odd primes $p$ we determine the fiber functors on $\mathrm{Rep}(H_p)$, where $H_p$ denotes Nikshych's semisimple Hopf algebra of dimension $4p^2$: there is one equivalence class if $p\equiv 3\pmod 4$ and two if $p\equiv 1\pmod 4$. We also use the classification for gaugings to determine which non-pointed entries in the small-dimensional list of Green and Nikshych are representation categories of semisimple factorizable Hopf algebras.

math.QA↗
arXiv2026

A Complexity Dichotomy for Quantum Invariants of 3-Manifolds

We determine the complexity of exact evaluation of the Reshetikhin--Turaev and Turaev--Viro invariants of closed connected oriented 3-manifolds, with the underlying tensor category fixed. If $\mathcal C$ is a modular category, then the Reshetikhin--Turaev invariant $Z_{\mathcal C}(M)$ can be computed in polynomial time from a framed-link surgery presentation of $M$ precisely when $\mathcal C$ is pointed; otherwise the problem is $\#\mathrm{P}$-hard. If $\mathcal A$ is a spherical fusion category, then the Turaev--Viro invariant $|M|_{\mathcal A}$ can be computed in polynomial time from a triangulation of $M$ precisely when the Drinfeld center $\mathcal Z(\mathcal A)$ is pointed, equivalently when $\mathcal A$ is trivializable pointed; otherwise the problem is $\#\mathrm{P}$-hard. This proves the dichotomy conjectured by Bridges and Samperton and identifies the categorical obstruction to polynomial-time evaluation.

math.QA↗
arXiv2026

Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

We show that single-qudit universality in Clifford-based gate sets follows a trichotomy determined by the prime factorization of the local dimension $d$. For prime $d$, any gate outside the Clifford group is universal. For prime-power dimensions $d=p^m$ with $m\ge 2$, not every non-Clifford gate is universal, but it can be achieved by suitable members of a family of diagonal phase gates, generalizing the qubit $T$ gate, as well as by permutations as simple as swapping $|0\rangle$ and $|1\rangle$ while leaving all other basis states unchanged. When $d$ decomposes into pairwise coprime prime powers, generalized CNOT-type gates between the corresponding factors already suffice for universality. In this composite case, universality can be obtained without introducing an explicit diagonal magic gate. Our results split non-Clifford resources for high-dimensional systems into two broad mechanisms: CNOT-type (permutations) or $T$-type (diagonal phases) gates.

quant-ph↗
arXiv2026

Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups

We explicitly construct nondegenerate braided $\mathbb{Z}_2$-crossed tensor categories of the form $\operatorname{Vect}_Γ\oplus\operatorname{Vect}_{Γ/2Γ}$. They are $\mathbb{Z}_2$-crossed extensions, in the sense of arXiv:0909.3140, of the braided tensor category $\operatorname{Vect}_Γ$ with $\mathbb{Z}_2$-action given by $-\mathrm{id}$ on the finite, abelian group $Γ$. Thus, we obtain generalisations of the Tambara-Yamagami categories, where now the abelian group $Γ$ may have even order and the nontrivial sector $\operatorname{Vect}_{Γ/2Γ}$ more than one simple object. The idea for this construction comes from a physically motivated approach in arXiv:2409.16357 to construct $\mathbb{Z}_2$-crossed extensions of $\operatorname{Vect}_Γ$ for any $Γ$ from an infinite Tambara-Yamagami category $\operatorname{Vect}_{\mathbb{R}^d}\oplus\operatorname{Vect}$, which itself is not fully rigorously defined, and then using condensation from $\operatorname{Vect}_{\mathbb{R}^d}$ to $\operatorname{Vect}_Γ$, which we prove commutes with crossed extensions. The $\mathbb{Z}_2$-equivariantisation of $\operatorname{Vect}_Γ\oplus\operatorname{Vect}_{Γ/2Γ}$ yields new modular tensor categories, which correspond to the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$, as discussed in arXiv:2409.16357.

math.QA↗
arXiv2025

A Note on the Eilenberg-Mac Lane Isomorphism for Quadratic Forms

We give an elementary proof of the Eilenberg-Mac Lane trace isomorphism between the third 2-abelian cohomology group and quadratic forms. Our approach yields explicit constructions and we characterize when quadratic forms can be expressed as traces of bilinear forms for arbitrary coefficient groups.

math.CT↗
arXiv2025

Categorical Fermionic Actions and Minimal Modular Extensions

We define fermionic actions of finite super-groups on fermionic fusion categories and establish necessary and sufficient conditions for their existence. Our main result characterizes when a braided fusion category admits a minimal non-degenerate extension in terms of cohomological obstructions. This characterization for braided fusion categories with non-Tannakian Müger center involves the fermionic structures and fermionic actions introduced in this work.

math.QA↗
arXiv2025

Zestings of Hopf Algebras

We extend the previously established zesting techniques from fusion categories to general tensor categories. In particular we consider the category of comodules over a Hopf algebra, providing a detailed translation of the categorical zesting construction into explicit Hopf algebraic terms: we show that the associative zesting of the category of comodules yields a coquasi-Hopf algebra whose comodule category is precisely the zested category. We explicitly write the modified multiplication and the associator, as well as the structures involved in the braided case. For pointed Hopf algebras, we derive concrete formulas for constructing zestings and establish a systematic approach for cyclic group gradings, providing explicit parameterizations of the zesting data.

math.QA↗
arXiv2025

Braided zesting and its applications

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

math.QA↗
arXiv2025

Twisted Kitaev quantum double model as local topological order

We study the Twisted Kitaev Quantum Double model within the framework of Local Topological Order (LTO). We extend its definition to arbitrary 2D lattices, enabling an explicit characterization of the ground state space through the invariant spaces of monomial representations. We reformulate the LTO conditions to include general lattices and prove that the twisted model satisfies all four LTO axioms on any 2D lattice. As a corollary, we show that its ground state space is a quantum error-correcting code.

math.QA↗
arXiv2024

The Condensed Fiber Product and Zesting

We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting. Furthermore, the condensed fiber product defines a monoidal structure on the 2-category of $G$-crossed and braided extensions of braided categories with a fixed transparent symmetric pointed subcategory.

math.QA↗
arXiv2024

Computing $G$-Crossed Extensions and Orbifolds of Vertex Operator Algebras

In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories. Their equivariantisations appear as categories of modules of fixed-point subalgebras (or orbifolds) of vertex operator algebras and are often difficult to determine. As the first tool, we show how the seminal work of Etingof, Nikshych and Ostrik on the uniqueness of $G$-crossed extensions can be used to determine the category of modules of orbifold vertex operator algebras. As an application, we determine the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$ for a lattice with odd-order discriminant form. In that case, the de-equivariantisation is of Tambara-Yamagami type. As the second tool, we describe how $G$-crossed extensions and condensations by commutative algebras commute in a suitable sense. This leads to an effective approach to compute new $G$-crossed extensions. As one application, we produce the coherence data that are then used in arXiv:2411.12251 to define a generalisation of the Tambara-Yamagami categories with more than one simple object in the twisted sector. This also yields the modular tensor category of the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$. Finally, we sketch how to categorically approach the general problem of lattice orbifolds under lifts of arbitrary lattice involutions.

math.QA↗
arXiv2024

A manifestly Morita-invariant construction of Turaev-Viro invariants

We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory $\mathbf{Mod}^{\mathrm{sph}}(\mathcal{A})$ of spherical module categories over a spherical fusion category $\mathcal{A}$. The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev-Viro invariant associated to $\mathcal{A}$, thereby proving the independence of the Turaev-Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin-Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in $\mathbf{Mod}^{\mathrm{sph}}(\mathcal{A})$ that we develop in this article. A central tool are Nakayama-twisted traces on pivotal bimodule categories which we study beyond semisimplicity.

math.QA↗