arXiv subjects

Simon Lentner

Publications and source records attributed to Simon Lentner.

At least 19 recordsLinked to original sources

arXiv2026

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

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arXiv2024

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 $\operatorname{Vect}_\Gamma\oplus\operatorname{Vect}_{\Gamma/2\Gamma}$. They are $\mathbb{Z}_2$-crossed extensions, in the sense of arXiv:0909.3140, of the braided tensor category $\operatorname{Vect}_\Gamma$ with $\mathbb{Z}_2$-action given by $-\mathrm{id}$ on the finite, abelian group $\Gamma$. Thus, we obtain generalisations of the Tambara-Yamagami categories, where now the abelian group $\Gamma$ may have even order and the nontrivial sector $\operatorname{Vect}_{\Gamma/2\Gamma}$ 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 $\operatorname{Vect}_\Gamma$ for any $\Gamma$ 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 $\operatorname{Vect}_\Gamma$, which we prove commutes with crossed extensions. The $\mathbb{Z}_2$-equivariantisation of $\operatorname{Vect}_\Gamma\oplus\operatorname{Vect}_{\Gamma/2\Gamma}$ 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.

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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.

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arXiv2023

Fibre functors and reconstruction of Hopf algebras

The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of representations of a braided Hopf algebra $H(F)$ in $C$. We prove that this is the case iff $B$ has the additional structure of a monoidal $C$-module category compatible with $F$, which equivalently means that $F$ admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fibre functors, and we give some applications. One particular motivation was the logarithmic Kazhdan-Lusztig conjecture.

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arXiv2023

An algebraic theory for logarithmic Kazhdan-Lusztig correspondences

Let $\mathcal{U}$ be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize $\mathcal{U}$ under the assumption that there exists a commutative algebra $A$ in $\mathcal{U}$ with certain properties: Let $\mathcal{C}$ be the category of local $A$-modules in $\mathcal{U}$ and $\mathcal{B}$ the category of $A$-modules in $\mathcal{U}$, which are in our set-up usually much simpler categories than $\mathcal{U}$. Then we can characterize $\mathcal{U}$ as a relative Drinfeld center $\mathcal{Z}_\mathcal{C}(\mathcal{B})$ and $\mathcal{B}$ as representations of a certain Hopf algebra inside $\mathcal{C}$. In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that $\mathcal{U}$ is abelian equivalent to the category of modules of some quantum group $U_q$ or some generalization thereof, and if $\mathcal{C}$ is braided equivalent to a category of graded vector spaces, and if $A$ has a certain form, then we already obtain a braided tensor equivalence between $\mathcal{U}$ and $\mathrm{Rep}(U_q)$. A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra $A$ and the corresponding category $\mathcal{C}$ are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra $\mathcal{M}(p)$ and the unrolled small quantum groups of $\mathfrak{sl}_2$ at $2p$-th root of unity. Another new example is the Kazhdan-Lusztig correspondence for $\mathfrak{gl}_{1|1}$.

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arXiv2022

Pointed Hopf algebras over non-abelian groups with non-simple standard braidings

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with non-simple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch--Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over non-abelian groups by Heckenberger--Vendramin, which consist of low rank exceptions and large rank families. We prove that the large rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large rank families, prove generation in degree one and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.

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arXiv2021

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fact that a vertex operator algebra and its gradewise dual need not be isomorphic as modules and that the tensor product of modules over vertex operator algebras need not be exact. We develop criteria which allow the detection of ribbon Grothendieck-Verdier equivalences and use them to explore ribbon Grothendieck-Verdier structures in the example of the rank $n$ Heisenberg vertex operator algebra or chiral free boson on a not necessarily full rank even lattice with arbitrary choice of conformal vector. We show that these categories are equivalent, as ribbon Grothendieck-Verdier categories, to certain categories of graded vector spaces and categories of modules over a certain Hopf algebra.

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arXiv2021

Characterizing braided tensor categories associated to logarithmic vertex operator algebras

Given a non-semisimple braided tensor category, with oplax tensor functors from known braided tensor categories, we ask : How does this knowledge characterize the tensor product and the braiding? We develop tools that address this question. In particular we prove that the associator is fixed by the oplax tensor functors, and we show that a distinguished role is played by the coalgebra structure on the image of theses tensor functors. Our setup constrains the form of quasi bialgebras appearing in the logarithmic Kazhdan-Lusztig conjecture and it applies in particular to the representation categories of the triplet vertex algebras. Here the two oplax tensor functors are determined by two free field realizations, and the coalgebras mentioned above are the Nichols algebras of type $\mathfrak{sl}_2$. We demonstrate in the case of $p=2$ that our setup completely determines the braided tensor category and the realizing quasi-triangular quasi-Hopf algebra is as anticipated in \cite{FGR2}. This proves the logarithmic Kazhdan-Lusztig conjecture for $p=2$, while for general $p$ it only remains to establish that our characterization provides a unique braided tensor category.

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arXiv2020

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.

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arXiv2018

Modularization of small quantum groups

We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for quantum groups at even roots of unity, where typically the initial representation category is not even braided. Our quasi-Hopf algebras are built from modules over the twisted Drinfeld double via a universal construction, but we also work out explicit generators and relations, and we prove that these algebras are modularizations of the quantum group extensions with R-matrices listed in [LO17]. As an application, we find one distinguished factorizable quasi-Hopf algebra for any finite root system and any root of unity of even order (resp. divisible by 4 or 6, depending on the root length). Under the same divisibility condition on a rescaled root lattice, a corresponding lattice Vertex-Operator Algebra contains a VOA W defined as the kernel of screening operators. We then conjecture that W representation categories are braided equivalent to the representation categories of the distinguished factorizable quasi-Hopf algebras. For A_1 root system, our construction specializes to the quasi-Hopf algebras in [GR17b, CGR17], where the answer is affirmative, similiary for B_n at fourth root of unity in [FGR17b, FL17].

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arXiv2017

Hochschild Cohomology and the Modular Group

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.

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arXiv2016

Factorizable $R$-Matrices for Small Quantum Groups

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig to endow these categories with the structure of a braided tensor category. In this article we determine all solutions to this ansatz that lead to a non-degenerate braiding. Particularly interesting are cases where the order of $q$ has common divisors with root lengths. In this way we produce familiar and unfamiliar series of (non-semisimple) modular tensor categories. In the degenerate cases we determine the group of so-called transparent objects for further use.

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arXiv2016

A decomposition of the Brauer-Picard group of the representation category of a finite group

We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra $H$ and thus the Brauer-Picard group of $H$-$\mathrm{mod}$. We consider two natural subgroups and a subset as candidates for generators. In this article $H$ is the group algebra of a finite group $G$. As our main result we prove that any element of the Brauer-Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups $G$ our decomposition reduces to the Bruhat decomposition of the Brauer-Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in $3d$-TQFT and group extensions of fusion categories.

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arXiv2016

A simplicial complex of Nichols algebras

We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for Nichols algebras over finite groups and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.

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arXiv2015

On monoidal autoequivalences of the category of Yetter-Drinfeld modules over a group: The lazy case

An interesting open question is to determine the group of monoidal autoequivalences of the category of Yetter-Drinfeld modules over a finite group $G$, or equivalently the group of Bigalois objects over the dual of the Drinfeld double $DG$. In particular one would hope to decompose this group into terms related to monoidal autoequivalences for the group algebra, the dual group algebra and interaction terms. We report on our progress in this question: We first prove a decomposition of the group of Hopf algebra automorphisms of the Drinfeld double into three subgroups, which reduces in the case $G=\mathbb{Z}_p^n$ to a Bruhat decomposition of $\mathrm{GL}_{2n}(\mathbb{Z}_p)$. Secondly, we propose a Kuenneth-like formula for the Hopf algebra cohomology of $DG^*$ into three terms and prove partial results in the case of lazy cohomology. We use these results for the calculation of the Brauer-Picard group in the lazy case in [LP15].

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arXiv2015

New $R$-matrices for small quantum groups

In this article we construct a large family of $R$-matrices for various extensions of small quantum groups by grouplike elements. The extensions are in correspondence to lattices between root and weight lattice and admit $R$-matrices in many cases where the original small quantum group does not. The results of this work extends some results that are well-known for $u_q(\mathfrak{sl}_2)$. Especially for $A_n$ a combinatorial theorem about roots of unity was necessary, which is interesting on its own, and has been proven in a separate paper [LN14].

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arXiv2014

A theorem on roots of unity and a combinatorial principle

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the non-negativity of certain alternating sums is equivalent to the set system being a filter. As an application we determine all discrete Fourier pairs of $\{0,1\}$-matrices. This technical result is an essential step in the classification of $R$-matrices of quantum groups.

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arXiv2014

Factorization of Graded Traces on Nichols Algebras

We study the factorization of the Hilbert series and more general graded traces of a Nichols algebra into cyclotomic polynomials. Nichols algebras play the role of Borel subalgebras of finite-dimensional quantum groups, so this observation can be viewed as an analog to the factorization of the order of finite Lie groups into cyclotomic polynomials. We prove results on this factorization and give a table of many examples of rank 1 over nonabelian groups.

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