arXiv subjects

Simon D. Lentner

Publications and source records attributed to Simon D. Lentner.

16 recordsLinked to original sources

arXiv2026

Lecture notes on Nichols algebras

These are lecture notes for an introductory course on Nichols algebras. As a main reference, I work with the book by Heckenberger and Schneider, but I want to take a distinct categorical perspective and try to develop the topic for an audience without a background in Hopf algebras. On the other hand I put some emphasis on hands-on examples. My first goal is to explain the definitions and the striking properties of Nichols algebras, foremost the odd reflection theory that is already present in Lie superalgebras. My second goal is to explain how the category of representations of a quantum group can be constructed, using categorical tools, from the Nichols algebra as its centerpiece. This makes the zoo of different existing versions of quantum groups more transparent and allows the construction of many more non-semisimple modular tensor categories. Other topics include different types of examples beyond the diagonal case, categorical versions of some Hopf algebra constructions, and an outlook section on the appearance of Nichols algebras in conformal field theory.

math.QA↗

arXiv2025

Nichols algebras, tensor categories and Kazhdan-Lusztig correspondences

There is a very general picture emerging that conjecturally describes what happens to the representation theory of a vertex algebra $\mathcal{V}$ if we pass to the kernel $\mathcal{W}$ of a set of screening operators. Namely, the screening operators generate a Nichols algebra $H$ inside $\mathrm{Rep}(\mathcal{V})$ and in many cases $\mathrm{Rep}(\mathcal{W})$ coincides with the relative Drinfeld center of $\mathrm{Rep}(H)$. This vastly generalizes the construction of a quantum group as the Drinfeld double of a Nichols algebra over the Cartan part. In this example, the conjectural category equivalence has been studied since around $20$ years as logarithmic Kazhdan Lusztig correspondence. The present survey was part of my habilitation thesis about my work in this area. I want to make it available as an introductory text, intended for readers from a pure algebra background as well as from a physics background. I motivate and explain gently and informally the different topics involved (quantum groups, Nichols algebras, vertex algebras, braided tensor categories) with a distinct categorical point of view, to the point that I can explain my general expectation. Then I explain some previous results and explain the main techniques in my recent proof of the conjectured category equivalence in case $\mathcal{V}$ is a free field theory and under technical assumptions on the analysis side.

math.QA↗

arXiv2025

Coupling a vertex algebra to a large center

Suppose a Lie group $G$ acts on a vertex algebra $V$. In this article we construct a vertex algebra $\tilde{V}$, which is an extension of $V$ by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $\mathfrak{g}$-connections $(d+A)$. The category of representations of $\tilde{V}$ fibres over the set of connections, and the fibres should be viewed as $(d+A)$-twisted modules of $V$, generalizing the familiar notion of $g$-twisted modules. In fact, another application of our result is that it proposes an explicit definition of $(d+A)$-twisted modules of $V$ in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. I particular we have in mind limits of the generalized quantum Langlands kernel, in which case $G$ is the Langland dual and $V$ is conjecturally the Feigin-Tipunin vertex algebra and the extension $\tilde{V}$ is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.

math.QA↗

arXiv2025

A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such an equivalence, and more general versions, using mainly algebraic arguments that characterize the representation category of the quantum group by quantities that are accessible on the vertex algebra side. Our proof is conditional on suitable analytic properties of the vertex algebra and its representation category. More precisely, we assume that it is a finite braided rigid monoidal category where the Frobenius-Perron dimensions are given by asymptotics of analytic characters.

math.QA↗

arXiv2024

Vertex algebras with big center and a Kazhdan-Lusztig Correspondence

Certain deformable families of vertex algebras acquire at a limit of the deformation parameter a large center, similar to affine Lie algebras at critical level. Then the vertex algebra and its representation category become a bundle over the variety defined by this large center. The zero-fibre becomes a vertex algebra, the other fibres become twisted modules over this vertex algebra. We explore these ideas and a conjectural correspondence in a class of vertex algebras $A^{(p)}[\mathfrak{g},κ]$ associated to a choice of a finite-dimensional semisimple Lie algebra and an integer $(\mathfrak{g},p)$ and a level $κ$, and some related algebras. These algebras were introduced under the name quantum geometric Langlands kernels and have an interpretation in 4-dimensional quantum field theory. In the limit $κ\to \infty$, they acquire a large central subalgebra identified with the ring of functions on the space of $\mathfrak{g}$-connections. The zero fibre is expected to be the Feigin-Tipunin algebra $W_p(\mathfrak{g})$, whose category of representations is expected to be equivalent to the small quantum group by the logarithmic Kazhdan Lusztig conjecture. Our rigorous results focus on the cases $(\mathfrak{g},1)$ and $(\mathfrak{sl}_2,2)$. In the first part of the paper, which might be of independent interest, we discuss the twisted modules (including twists by unipotent elements) of the affine Lie algebra and of the triplet algebra $W_2(\mathfrak{sl}_2)$, and introduce a method of twisted free field realization to obtain twisted modules for arbitrary $(\mathfrak{g},p)$. We also match our findings to the respective quantum group with big center. In the second part of the paper we match the results to the vertex algebra bundle we compute from the limit of $A^{(p)}[\mathfrak{g},κ]$, which in our cases have realizations as GKO coset resp. N=4 superconformal algebras.

hep-th↗

arXiv2024

Twisted vertex algebra modules for irregular connections: A case study

A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\mathrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disc. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions, or equivalently the triplet vertex algebra $\mathcal{W}_p(\mathfrak{sl}_2)$ for $p=2$, and study its twisted module with respect to irregular $\mathfrak{sl}_2$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, and in particular in work by Witten, higher dimensional field theories and AGT correspondence. Our original motivation comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections [FL24], similar to the affine Lie algebra at critical level. Our present article now describes, in the smallest case, the fibres of this category over irregular connections.

math.QA↗

arXiv2024

On Borel subalgebras of quantum groups

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of $U_q(\mathfrak{sl}_2)$ and $U_q(\mathfrak{sl}_3)$ and discuss the induced modules.

math.QA↗

arXiv2022

Algebras of Non-Local Screenings and Diagonal Nichols Algebras

In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated to a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center.

math.QA↗

arXiv2021

Quantum groups and Nichols algebras acting on conformal field theories

We prove that certain screening operators in conformal field theory obey the algebra relations of a corresponding Nichols algebra with diagonal braiding. Our result proves in particular a long-standing expectation that the Borel parts of small quantum groups appear as the algebra of screening operators. The proof is based on a novel, intimate relation between Hopf algebras, vertex algebras and a class of multivalued analytic special functions, which are generalizations of Selberg integrals. We prove that the zeroes of these special functions correspond to the algebra relations of the respective Nichols algebra, by proving an analytical quantum symmetrizer formula for the functions. Moreover, certain poles of the functions encode module extensions and a Weyl group action. At other poles, the quantum {symmetrizer} formula fails and the screening operators generate an extension of the Nichols algebra. The intended application of our result is the conjectural logarithmic Kazhdan-Lusztig correspondence. More generally, our result seems to suggest that non-local screening operators in an arbitrary vertex algebra should be described by appropriate Nichols algebras, just as local screening operators can be described by Lie algebras.

math.QA↗

arXiv2019

A Frobenius homomorphism for Lusztig's quantum groups over arbitrary roots of unity

For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots, and may produce Lie algebras in a symmetrically braided category.

math.RT↗

arXiv2019

The unrolled quantum group inside Lusztig's quantum group of divided powers

In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig's quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct adjoint action. As application we explain how this gives a realization of the unrolled quantum group as operators on a conformal feld theory and match some calculations on this side. In particular our results explain a prominent weight shift that appears in [FT10]. Our result extends to other Nichols algebras of diagonal type, including super Lie algebras.

math.QA↗

arXiv2017

Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.

math.QA↗

arXiv2015

New Large-Rank Nichols Algebras Over Nonabelian Groups With Commutator Subgroup Z_2

In this article, we explicitly construct new finite-dimensional, link-indecomposable Nichols algebras with Dynkin diagrams of type An,Cn,Dn,E6,E7,E8,F4 over any group G with commutator subgroup isomorphic to Z_2.The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomory in 1995 [Mont95][AS02]. Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [Len12] and produces non-faithful Nichols algebras. However, we give faithful examples of Doi twists for type A3,C3,D4,F4 over several nonabelian groups of order 16 and 32. These are hence the first known examples of faithful, finite-dimensional, link-indecomposable Nichols algebras of rank >2 over nonabelian groups.

math.QA↗

arXiv2015

Nash Equilibria And Partition Functions Of Games With Many Dependent Players

We discuss and solve a model for a game with many players, where a subset of truely deciding players is embedded into a hierarchy of dependent agents. These interdependencies modify the game matrix and the Nash equilibria for the deciding players. In a concrete example, we recognize the partition function of the Ising model and for high dependency we observe a phase transition to a new Nash equilibrium, which is the Pareto-efficient outcome. An example we have in mind is the game theory for major shareholders in a stock market, where intermediate companies decide according to a majority vote of their owners and compete for the final profit. In our model, these interdependency eventually forces cooperation.

cs.GT↗

arXiv2014

Quantum affine algebras at small root of unity

We study the Frobenius-Lusztig kernel for quantum affine algebras at root of unity of small orders that are usually excluded in literature. These cases are somewhat degenerate and we find that the kernel is in fact mostly related to different affine Lie algebras, some even of larger rank, that exceptionally sit inside the quantum affine algebra. This continues the authors study for quantum groups associated to finite-dimensional Lie algebras in [Len14c].

math.QA↗

arXiv2014

Root Systems In Finite Symplectic Vector Spaces

We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify finite-dimensional Nichols algebras for nilpotent groups G over the complex numbers [Len13a], where the symplectic form is given by the group's commutator map. For example, the degree of degeneracy of the symplectic root system determines the size of the center of G. In this article we classify symplectic root systems over the finite field F_2, where symplectic just means isotropic. We prove that every Dynkin diagram admits, up to symplectic isomorphisms, a unique minimal symplectic root system over F_2 and thus requires a specific degree of degeneracy of the symplectic vector space. Any non-minimal symplectic root system turns out to be a quotient of a minimal one by a universal property. As examples and for further applications we explicitly construct all symplectic root systems for Cartan matrices resp. Dynkin diagrams of type ADE.

math.QA↗