Finite-dimensional Nichols Algebras Research Articles

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.

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.

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

Lie Algebras Arising from Nichols Algebras of Diagonal Type

Abstract Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in [ 4], and $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in [ 5]. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$, which is determined for each $\mathfrak{q}$ in the list of [ 25].

Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

AbstractWe present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

Finite-dimensional Hopf algebras over the Hopf algebra Hb:1 of Kashina

Let H be the 16-dimensional nontrivial semisimple Hopf algebra Hb:1 appeared in Kashina's work [32]. We obtain all simple Yetter-Drinfeld modules over H and then determine all finite-dimensional Nichols algebras satisfying B(V)≅⨂i∈IB(Vi), where V=⨁i∈IVi, each Vi is a simple object in YDHH. Finally, we describe all liftings of those B(V) over H.

Quantum function algebras from finite-dimensional Nichols algebras

We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra B, our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H-comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them quantum function algebras from Fomin-Kirillov algebras associated with the symmetric groups on three letters.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from mathbf {PSL}_n(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is mathbf {PSp}_{2n}(q), mathbf {P}{varvec{Omega }}^+_{4n}(q), mathbf {P}{varvec{Omega }}^-_{4n}(q), ^3D_4(q), E_7(q), E_8(q), F_4(q), or G_2(q) with q even is the group algebra.

Finite-dimensional Nichols algebras over dual Radford algebras

For [Formula: see text], let [Formula: see text] be the dual of the Radford algebra of dimension [Formula: see text]. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over [Formula: see text]. Along the way, we describe the simple objects in [Formula: see text] and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case [Formula: see text]. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for [Formula: see text] and [Formula: see text], [Formula: see text], which recovers some results of the second and third author in the former case, and of Xiong in the latter.Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es.Jorge Luis Borges

On the restricted Jordan plane in odd characteristic

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

Finite-dimensional Hopf algebras over the Hopf algebra H:−1,1 of Kashina

Let H be the 16-dimensional non-trivial semisimple Hopf algebra H d : − 1 , 1 in the classification work of Kashina [28] . Actually, H ≅ H 8 ⊗ k C 2 , where H 8 is the Kac-Paljutkin algebra [26] . Let V = ⨁ i ∈ I V i , where V i is a simple object in YD H H . All finite-dimensional Nichols algebras satisfying B ( V ) ≅ ⨂ i ∈ I B ( V i ) are determined completely. Under this assumption, we consider and classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra H d : − 1 , 1 via the relevant Nichols algebras B ( V ) .

Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In the present paper, all rank 4 Nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified. Our proof uses the classification of the finite arithmetic root systems of rank 4.

Finite dimensional Hopf algebras over the Kac-Paljutkin algebra $H_8$.

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method.

Finite-Dimensional Pointed Hopf Algebras Over Finite Simple Groups of Lie Type IV. Unipotent Classes in Chevalley and Steinberg Groups

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSLn(q) and PSp2n(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSUn(2m).

Factorization of Graded Traces on Nichols Algebras

A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize.

ON PROJECTIVE MODULES OVER FINITE QUANTUM GROUPS

Let $\mathcal{D}$ be the Drinfeld double of the bosonization ${\mathfrak B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${\mathfrak B}(V)$ over a finite group $G$. It is known that the simple $\mathcal{D}$-modules are parametrized by the simple modules over $\mathcal{D}(G)$, the Drinfeld double of $G$. This parametrization can be obtained by considering the head $\mathsf{L}(\lambda)$ of the Verma module $\mathsf{M}(\lambda)$ for every simple $\mathcal{D}(G)$-module $\lambda$. In the present work, we show that the projective $\mathcal{D}$-modules are filtered by Verma modules and the BGG Reciprocity $[\mathsf{P}(\mu):\mathsf{M}(\lambda)]=[\mathsf{M}(\lambda):\mathsf{L}(\mu)]$ holds for the projective cover $\mathsf{P}(\mu)$ of $\mathsf{L}(\mu)$. We use graded characters to proof the BGG Reciprocity and obtain a graded version of it. Also, we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.

Examples of finite-dimensional Hopf algebras with the dual Chevalley property

We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra.