Nichols Algebras Research Articles

Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.

Some classifications of finite-dimensional Hopf algebras over Ha:y of Kashina

Let [Formula: see text] be the [Formula: see text]-dimensional nontrivial semisimple Hopf algebra [Formula: see text] appeared in Kashina’s work [Classification of semisimple Hopf algebras of dimension 16, J. Algebra 232(2) (2000) 617–663]. We obtain all simple Yetter–Drinfeld modules over [Formula: see text] and then determine all finite-dimensional Nichols algebras satisfying [Formula: see text], where [Formula: see text], each [Formula: see text] is a simple object in [Formula: see text]. Moreover, for Nichols algebras satisfying [Formula: see text], we list some infinite dimensional Nichols algebras [Formula: see text] and obtain finite dimensional Nichols algebras of diagonal type that are either super type or [Formula: see text]. Finally, we describe some liftings of those [Formula: see text] over [Formula: see text].

On Hopf algebras whose coradical is a cocentral abelian cleft extension

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn , n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K 3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216.

Bosonization of curved Lie bialgebras

We use Cartier's preadditive symmetric monoidal categories to study Lie bialgebras. We prove that bosonization can be done consistently in this framework. In the last part of the paper we present explicit examples and indicate a deep relationship between certain curved Lie bialgebras and Nichols algebras over abelian groups.

A characterization of Nichols algebras of diagonal type which are free algebras

A characterization of Nichols algebras of diagonal type which are free algebras

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSLn(q) and PSp2n(q)

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSLn(q)(n≥4), PSL3(q)(q>2), or PSp2n(q)(n≥3), is isomorphic to a group algebra.

Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

AbstractWe 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 nonabelian group with nonsimple 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 nonabelian 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 1, 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.

On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type

On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type

Finite-dimensional Nichols algebras over the Suzuki algebras II: Simple Yetter–Drinfeld modules of AN2n+1μλ

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra [Formula: see text] and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type [Formula: see text], [Formula: see text], [Formula: see text], Super type [Formula: see text] and the Nichols algebra [Formula: see text]. And the involved finite-dimensional Nichols algebras of non-diagonal type are [Formula: see text], [Formula: see text] and [Formula: see text]-dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over [Formula: see text].

Rank 4 Nichols Algebras of Pale Braidings

We classify finite GK-dimensional Nichols algebras B(V) of rank 4 such that V arises as a Yetter-Drinfeld module over an abelian group but it is not a direct sum of points and blocks.

Higher Braidings of Diagonal Type

Heckenberger introduced the Weyl groupoid of a finite-dimensional Nichols algebra of diagonal type. We replace the matrix of its braiding by a higher tensor and present a construction which yields further Weyl groupoids. Abelian cohomology theory gives evidence for the existence of a higher braiding associated to such a tensor.

Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type

We show that every finite GK-dimensional pre-Nichols algebra for braidings of diagonal type with connected diagram of modular, supermodular or unidentified type is a quotient of the distinguished pre-Nichols algebra introduced by the first-named author, up to two exceptions. For these two exceptional cases, we provide a pre-Nichols algebra that substitutes the distinguished one in the sense that it projects onto all finite GK-dimensional pre-Nichols algebras. We build both substitutes as non-trivial central extensions with finite GK-dimension of the corresponding distinguished pre-Nichols algebra. We describe these algebras by generators and relations and give a basis. This work essentially completes the study of eminent pre-Nichols algebras of diagonal type with connected diagram and finite-dimensional Nichols algebra.

Nichols Algebras and Quantum Principal Bundles

Abstract We introduce a general framework for associating to a homogeneous quantum principal bundle a Yetter–Drinfeld module structure on the cotangent space of the base calculus. The holomorphic and anti-holomorphic Heckenberger–Kolb calculi of the quantum Grassmannians are then presented in this framework. This allows us to express the calculi in terms of the corresponding Nichols algebras. The extension of this result to all irreducible quantum flag manifolds is then conjectured.

Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation

In this paper, we investigate the Nichols algebra associated to any non-degenerate involutive solution (X, r) of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension nm with m, . It turns out that the Nichols algebra has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal’s triangle.

On the Laistrygonian Nichols algebras that are domains

We consider a class of Nichols algebras $$\mathscr {B}(\mathfrak L_q( 1, \mathscr {G}))$$ introduced in Andruskiewitsch et al. which are domains and have many favourable properties like AS-regular and strongly noetherian. We classify their finite-dimensional simple modules and their point modules.

Finite-dimensional Nichols algebras over the Suzuki algebras I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{\mu\lambda}$

The Suzuki algebra $A_{Nn}^{\mu \lambda}$, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;\I_2)$ and the Nichols algebra $\ufo(8)$. There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. Using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

Classifying Nichols algebras over classical Weyl groups

In this paper, we show that conjugacy classes of classical Weyl groups [Formula: see text] and [Formula: see text] are of type D. Consequently, we obtain that Nichols algebras of irreducible Yetter–Drinfeld modules over the classical Weyl groups [Formula: see text] ([Formula: see text]) are infinite dimensional.

On some classification of finite-dimensional Hopf algebras over the Hopf algebra of Kashina

Let H be the dual of 16-dimensional nontrivial semisimple Hopf algebra in the classification work of Kashina [22]. We completely determine all finite-dimensional Nichols algebras satisfying where each Ni is a simple object in Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra H via the relevant Nichols algebras

On the computation of Hopf 2-cocycles with an example of diagonal type

AbstractWe present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.

Geometric perspective on Nichols algebras

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category C. We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category C is the category YDGG of Yetter-Drinfeld modules over a finite group G, this reduces the question of generation by group-like elements and skew-primitives to a geometric question about rigidity of orbits. Comparing the results of Angiono Kochetov and Mastnak, this gives a new proof for the generation of finite dimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if V is a simple object in C and B(V) is finite dimensional, then B(V) must be rigid. We also show that a non-rigid Nichols algebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra which is isomorphic to the Nichols algebra as an object of the category C.