arXiv subjects

Harshit Yadav

Publications and source records attributed to Harshit Yadav.

15 recordsLinked to original sources

arXiv2026

A Galois connection between subalgebras and tensor subcategories

Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of $\mathcal{B}_A$. Let $\mathcal{B}'$ denote the M\"uger center and set $A':=A\cap\mathcal{B}'$. The closure operators are $B\mapsto\langle B,A'\rangle_{\mathrm{alg}}$ and $\mathcal{E}\mapsto\langle\mathcal{E},\mathcal{B}_A^{\mathrm{loc}}\rangle_{\otimes}$. Thus the closed subalgebras are those containing $A'$, while the closed tensor subcategories are those containing $\mathcal{B}_A^{\mathrm{loc}}$; equivalently, the fixed-point intervals $[A',A]_{\mathrm{alg}}$ and $[\mathcal{B}_A^{\mathrm{loc}},\mathcal{B}_A]_{\otimes}$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.

math.QA↗

arXiv2026

Transparent Subalgebras and Local Module Categories

Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the M\"uger center of the category of local $A$-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on $\mathrm{FPdim}_{\mathcal{B}}(A)$. We also realize the M\"uger center of $\mathcal{B}$ as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.

math.QA↗

arXiv2026

Ribbon categories from ind-exact algebras: simple current case

We give criteria for when finitely generated local modules over a commutative algebra $A$ in the ind-completion $\widehat{\mathcal{C}}$ of a braided tensor category $\mathcal{C}$ inherit the structure of a (rigid, braided, ribbon) tensor category. We then apply this to simple current algebras $A = \bigoplus_{g \in \Gamma} E_g$, where $\Gamma$ is a subgroup of invertible objects in $\mathcal{C}$. Using a description of simple $A$-modules, we verify the required hypotheses for this class of algebras and deduce rigidity, braided, ribbon, and non-degeneracy properties for their finitely generated local modules. As applications, we construct examples of ribbon tensor categories from quantum supergroup categories for unrolled $\mathfrak{gl}(1|1)$.

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arXiv2025

Pivotal Brauer-Picard groupoids and graded extensions

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ $2$-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal $2$-functors into the pivotal Brauer-Picard $2$-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.

math.QA↗

arXiv2025

$\otimes$-Frobenius functors and exact module categories

We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion -- most notably, $F$ is $\otimes$-Frobenius if and only if the centralizer $Z({}_{F}\!{\mathcal{D}}_{\!F})$ is unimodular. We use them to analyze how actions on module categories behave under pullback along $F$. For perfect functors, we show that twisting a $\mathcal{D}$-module category $\mathcal{M}$ along $F$ preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever $F$ is $\otimes$-Frobenius (or, more generally, Frobenius with respect to $\mathcal{M}$). Applications include: (i) explicit criteria for $\otimes$-Frobenius functors arising from bialgebra maps $f\!:\!H'\!\to\!H$ between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in $Z(\mathcal{C})$. Along the way we show that central tensor functors are Frobenius iff they are $\otimes$-Frobenius and that any tensor functor between separable fusion categories is $\otimes$-Frobenius, answering questions of Flake-Laugwitz-Posur.

math.QA↗

arXiv2024

Nondegenerate module categories

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.

math.QA↗

arXiv2024

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ and its subcategory $\mathcal{C}_A^{\text{loc}}$ of local modules inherit rigidity from $\mathcal{C}$, and then we find conditions for $\mathcal{C}$ and $\mathcal{C}_A$ to inherit rigidity from $\mathcal{C}_A^{\text{loc}}$. First, we assume $\mathcal{C}$ is a braided finite tensor category and prove rigidity of $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ under conditions based on criteria of Etingof-Ostrik for $A$ to be an exact algebra in $\mathcal{C}$. As a corollary, we show that if $A$ is a simple $\mathbb{Z}_{\geq 0}$-graded VOA with a strongly rational vertex operator subalgebra $V$, then $A$ is strongly rational, without requiring the categorical dimension of $A$ as a $V$-module to be non-zero. Next, we assume $\mathcal{C}$ is a Grothendieck-Verdier category, i.e., $\mathcal{C}$ admits a weaker duality structure than rigidity. We first prove $\mathcal{C}_A$ is also a Grothendieck-Verdier category. Using this, we prove that if $\mathcal{C}_A^{\text{loc}}$ is rigid, then so is $\mathcal{C}$ under conditions such as a mild non-degeneracy assumption on $\mathcal{C}$, an assumption that every simple object of $\mathcal{C}_A$ is local, and that induction from $\mathcal{C}$ to $\mathcal{C}_A$ commutes with duality. These conditions are motivated by free field-like VOA extensions $V\subseteq A$ where $A$ is often an indecomposable $V$-module, so our result will make it more feasible to prove rigidity for many vertex algebraic monoidal categories. In a follow-up work, our result will be used to prove rigidity of the category of weight modules for the simple affine VOA of $\mathfrak{sl}_2$ at any admissible level.

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arXiv2024

Commutative exact algebras and modular tensor categories

Inspired by the study of vertex operator algebra extensions, we answer the question of when the category of local modules over a commutative exact algebra in a braided finite tensor category is a (non-semisimple) modular tensor category. Along the way we provide sufficient conditions for the category of local modules to be rigid, pivotal and ribbon. We also discuss two ways to construct such commutative exact algebras. The first is the class of simple current algebras and the second is using right adjoints of central tensor functors. Furthermore, we discuss Witt equivalence and its relation with extensions of VOAs.

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arXiv2023

On unimodular module categories

Let $\mathcal{C}$ be a finite tensor category and $\mathcal{M}$ an exact left $\mathcal{C}$-module category. We call $\mathcal{M}$ unimodular if the finite multitensor category ${\sf Rex}_{\mathcal{C}}(\mathcal{M})$ of right exact $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When $\mathcal{C}$ is a pivotal category, and $\mathcal{M}$ is a unimodular, pivotal left $\mathcal{C}$-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

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arXiv2023

On non-counital Frobenius algebras

A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped with a coassociative, counital comultiplication map $Δ$ that is an $A$-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map $Δ$ as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

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arXiv2023

Frobenius monoidal functors from (co)Hopf adjunctions

Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a symmetric Frobenius algebra in $\mathcal{C}$. As an application, we construct Frobenius monoidal functors going into the Drinfeld center $\mathcal{Z}(\mathcal{C})$, thereby producing Frobenius algebras in it.

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arXiv2022

Filtered Frobenius algebras in monoidal categories

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal{C}$ is represented by a Frobenius algebra in $\mathcal{C}$. Several directions for further investigation are also proposed.

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arXiv2021

Autoequivariant Network Search via Group Decomposition

Recent works show that group equivariance as an inductive bias improves neural network performance for both classification and generation. However, designing group-equivariant neural networks is challenging when the group of interest is large and is unknown. Moreover, inducing equivariance can significantly reduce the number of independent parameters in a network with fixed feature size, affecting its overall performance. We address these problems by proving a new group-theoretic result in the context of equivariant neural networks that shows that a network is equivariant to a large group if and only if it is equivariant to smaller groups from which it is constructed. Using this result, we design a novel fast group equivariant construction algorithm, and a deep Q-learning-based search algorithm in a reduced search space, yielding what we call autoequivariant networks (AENs). AENs find the right balance between equivariance and network size when tested on new benchmark datasets, G-MNIST and G-Fashion-MNIST, obtained via group transformations on MNIST and Fashion-MNIST respectively that we release. Extending these results to group convolutional neural networks, where we optimize between equivariances, augmentations, and network sizes, we find group equivariance to be the most dominating factor in all high-performing GCNNs on several datasets like CIFAR10, SVHN, RotMNIST, ASL, EMNIST, and KMNIST.

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arXiv2019

Equivariant cohomology, Schubert calculus, and edge labeled tableaux

This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.

math.CO↗

arXiv2019

The A.B.C.Ds of Schubert calculus

We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.

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