Yetter Drinfeld Modules Research Articles

Double wreath quasi-Hopf algebras

For a quasi-bialgebra H, we show that the category C of H-bimodules is duoidal and that the so called u-coaugmented bimonoids in C are exactly the quasi-bialgebras with a coalgebra projection. When H is a quasi-Hopf algebra with bijective antipode, we prove that the u-coaugmented bimonoids in C can also be described as what we will call double wreath quasi-Hopf algebras, objects determined by H and pre-bialgebras R within the category of Yetter-Drinfeld modules over H. A particular class of double wreath quasi-Hopf algebras is obtained by deforming with a 2-cocycle the multiplication of a Radford biproduct quasi-Hopf algebra. Other classes of this type are given by the symplectic fermion quasi-Hopf algebras.

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].

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

AbstractWe find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

An algebraic framework for the Drinfeld double based on infinite groupoids

In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D(A) and Yetter-Drinfeld modules over a weak algebraic quantum group A.

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.

Weak Hopf algebras, smash products and applications to adjoint-stable algebras

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

On the structure of irreducible Yetter-Drinfeld modules over $ D $

A class of algebras $ D(m, d, \xi) $ introduced by <sup>[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]</sup> were not pointed and generated by the coradical of $ D(m, d, \xi) $. Let $ D $ be the quotient of $ D(m, d, \xi) $ module the principle ideal $ (g^m-1) $. First, we describe all simple left modules of $ D $. Then, according to Radford's method, we construct the Yetter-Drinfeld module over $ D $ by the tensor product of a simple module of $ D $ and $ D $ itself. Hence, we find some simple left Yetter-Drinfeld modules over $ D $, and the relevant braidings are of a triangular type.

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.

Yetter-Drinfeld modules over Nichols systems: reflections, induced objects and maximal subobject

We study Yetter-Drinfeld modules over Nichols systems to generalize results about Verma modules in the represation theory of Lie algebras to Nichols algebras of group type. We construct and study reflections of such objects and obtain geometric properties of their support. In particular we study a subcategory obtained by inducing comodules of Nichols systems. Finally we study the maximal subobject and irreducibility of such Yetter-Drinfeld modules. To do so we introduce a special morphism that we name Shapovalov morphism, to generalize a polynomial called Shapovalov determinant. We will study its properties and behavior under reflections.

Unifying Radford’s biproducts over Hopf braces

Let [Formula: see text] be a braided Hopf brace, i.e. a Hopf brace in the category of Yetter–Drinfeld modules over a Hopf brace [Formula: see text]. In this note, we give the sufficient and necessary conditions which make Radford’s biproduct [Formula: see text] and [Formula: see text] become a Hopf brace.

The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n 2-dimensional Kac-Paljutkin Hopf algebras

As is well known, a class of -dimensional Kac-Paljutkin Hopf algebras was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H 8. In this paper, the classification of Yetter-Drinfeld modules over is given by Radford’s method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over is established, and the Grothendieck ring is described explicitly by generators and relations. Communicated by Julia Plavnik

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.

A decomposition Theorem for pointed braided Hopf algebras

A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfills a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be decomposed into a tensor product of these two. Often one considers braided Hopf algebras in a Yetter-Drinfeld category of an ordinary Hopf algebra. In this case the braided Hopf algebra and many interesting coideal subalgebras are in particular comodules. We extend the mentioned Theorem by proving that the decomposition is compatible with this comodule structure if the underlying ordinary Hopf algebra is cosemisimple.

Functors for Long dimodules and Yetter–Drinfeld modules in a weak setting

In this paper, for two weak Hopf monoids H and B with invertible antipode, we define a functor between the category of left-left H\otimes B -Yetter–Drinfeld modules and the category of H - B -Long dimodules. We also show that, if moreover H is quasitriangular and B is coquasitriangular, this functor is a retraction of the well-known injective functor between left-left H - B -Long dimodules and left-left H\otimes B -Yetter–Drinfeld modules.

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.

Projective class rings of a kind of category of Yetter-Drinfeld modules

<abstract><p>In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed $ 8m $-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.</p></abstract>

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 $m\geq 5$.

Structures of adjoint-stable algebras over factorizable Hopf algebras

For a quasi-triangular Hopf algebra (H, R), there is a notion of transmuted braided group HR of H introduced by Majid. The transmuted braided group HR is a Hopf algebra in the braided category . The R-adjoint-stable algebra associated with any simple left HR -comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in . In this note, we prove for a semisimple factorizable Hopf algebra (H, R) that any simple subcoalgebra of HR is H-stable and the R-adjoint-stable algebra for any simple left HR -comodule is anti-isomorphic to H. As an application, we characterize all irreducible Yetter-Drinfeld modules.

Centers of braided tensor categories

Let C be a finite braided multitensor category. Then the end B=∫X∈CX⊗X⁎ is natural a Hopf algebra in C. We show that the Drinfeld center of C is isomorphic to the category of left B-comodules in C, and the decomposition of B into a direct sum of indecomposable C-subcoalgebras leads to a decomposition of B-ComodC into a direct sum of indecomposable C-module subcategories.As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple and cosemisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.