Cocycle Deformations Research Articles

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.

Cocycle deformations for weak Hopf algebras

In this paper we introduce a theory of multiplication alteration by two-cocycles for weak Hopf algebras. We show that, just like it happens for Hopf algebras, if H a weak Hopf algebra and H? its weak Hopf algebra deformation by a 2-cocycle ?, there is a braided monoidal category equivalence between the categories of left-right Yetter-Drinfel?d modules HYDH and H?YDH?. As a consequence we get in this context that the category Rep(D(H)) of left modules over the Drinfel?d double D(H) can be identified with the category Rep(D(H?)) of left modules over the Drinfel?d double D(H?).

Multiparameter quantum groups at roots of unity

We address the study of multiparameter quantum groups (MpQGs) at roots of unity, namely quantum universal enveloping algebras $U\_\mathbf{q}(\mathfrak{g})$ depending on a matrix of parameters $\mathbf{q}=(q\_{ij}){i,j\in I}$. This is performed via the construction of quantum root vectors and suitable “integral forms” of $U\mathbf{q}(\mathfrak{g})$, a restricted one—generated by quantum divided powers and quantum binomial coefficients—and an unrestricted one—where quantum root vectors are suitably renormalized. The specializations at roots of unity of either form are the “MpQGs at roots of unity” we look for. In particular, we study special subalgebras and quotients of our MpQGs at roots of unity—namely, the multiparameter version of small quantum groups—and suitable associated quantum Frobenius morphisms, that link the MpQGs at roots of 1 with MpQGs at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion, often at the core of our strategy, is that every MpQG is actually a $2$-cocycle deformation of the algebra structure of (a lift of) the “canonical” one-parameter quantum group by Jimbo–Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $\mathbf{q}$, our quantum groups yield (through the choice of integral forms and their specializations) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.

Pseudo-Riemannian metrics on bicovariant bimodules

We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a bicovariant bimodule and its cocycle deformations are in one to one correspondence.

Yetter–Drinfeld category for the quasi-Turaev group coalgebra and cocycle deformation

Let [Formula: see text] be a group and [Formula: see text] a quasi-Turaev group coalgebra over [Formula: see text]. In this paper, we firstly construct the category [Formula: see text] of left-right Yetter–Drinfeld modules over [Formula: see text], generalizing both of the Yetter–Drinfeld modules over Hopf group coalgebra and quasi-Hopf algebra, and prove that this category is isomorphic to the center of the representation category of [Formula: see text]. Next, we prove that the full subcategory [Formula: see text] consisting of all finite dimensional Yetter–Drinfeld modules over [Formula: see text] is autonomous. Finally, when [Formula: see text] is reduced to a Turaev group coalgebra, we introduce the dual 2-cocycle [Formula: see text] of [Formula: see text], and dual cocycle deformation [Formula: see text], and show that [Formula: see text] is isomorphic to [Formula: see text] as braided monoidal category.

Twisted deformations vs. cocycle deformations for quantum groups

In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by [Formula: see text]-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in short QUEAs), for which we accordingly consider those arising from twisted deformations (in short TwQUEAs) and those arising from [Formula: see text]-cocycle deformations, usually called multiparameter QUEAs (in short MpQUEAs). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the “canonical”, well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEAs and of MpQUEAs — which, in Hopf algebra theoretical terms are naturally dual to each other — actually coincide; thus, that there exists in fact only one type of “pluriparametric deformation” for QUEAs. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

Covariant connections on bicovariant differential calculus

Given a bicovariant differential calculus (E,d) such that the braiding map is diagonalisable in a certain sense, the bimodule of two-tensors admits a direct sum decomposition into symmetric and anti-symmetric tensors. This is used to prove the existence of a bicovariant torsionless connection on E. Following Heckenberger and Schmüdgen, we study invariant metrics and the compatibility of covariant connections with such metrics. A sufficient condition for the existence and uniqueness of bicovariant Levi-Civita connections is derived. This condition is shown to hold for cocycle deformations of classical Lie groups.

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

Hopf cocycle deformations and invariant theory

For a given finite dimensional Hopf algebra H we describe the set of all equivalence classes of cocycle deformations of H as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the Universal Coefficients Theorem in the case of a group algebra, and we also give examples from other families of Hopf algebras, including dual group algebras and Bosonizations of Nichols algebras. In particular, we use the methods developed here to classify the cocycle deformations of a dual pointed Hopf algebra associated to the symmetric group on three letters. We also give an example of a cocycle deformation over a dual group algebra, which has only rational invariants, but which is not definable over the rational field. This differs from the case of group algebras, in which every 2-cocycle is equivalent to one which is definable by its invariants.

Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type V with a principal realization in the category of Yetter–Drinfeld modules of a cosemisimple Hopf algebra H and such that the Nichols algebra $$\mathfrak {B}(V)$$ is finite-dimensional, thus presented by a finite set $${{\mathcal {G}}}$$ of relations, we define a family of Hopf algebras $$\mathfrak {u}(\varvec{\lambda })$$ , $$\varvec{\lambda }\in \Bbbk ^{{{\mathcal {G}}}}$$ , which are cocycle deformations of $$\mathfrak {B}(V)\# H$$ and such that $${\text {gr}}\mathfrak {u}(\varvec{\lambda })\simeq \mathfrak {B}(V)\# H$$ .

On functor points of affine supergroups

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature.

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension p3 and of dimension pq2. We obtain that the p+1 non-isomorphic self-dual semisimple Hopf algebras of dimension p3 classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension p3, established by the third-named author in an appendix.

The Projective Class Rings of a family of pointed Hopf algebras of Rank two

In this paper, we compute the projective class rings of the tensor product $\mathcal{H}_n(q)=A_n(q)\ot A_n(q^{-1})$ of Taft algebras $A_n(q)$ and $A_n(q^{-1})$, and its cocycle deformations $H_n(0,q)$ and $H_n(1,q)$, where $n>2$ is a positive integer and $q$ is a primitive $n$-th root of unity. It is shown that the projective class rings $r_p(\mathcal{H}_n(q))$, $r_p(H_n(0,q))$ and $r_p(H_n(1,q))$ are commutative rings generated by three elements, three elements and two elements subject to some relations, respectively. It turns out that even $\mathcal{H}_n(q)$, $H_n(0,q)$ and $H_n(1,q)$ are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.

Deformations of spectral triples and their quantum isometry groups via monoidal equivalences

In this paper, we propose a new procedure to deform spectral triples and their quantum isometry groups. The deformation data are a spectral triple (mathcal {A},mathcal {H}, D), a compact quantum group {mathbb {G}} acting algebraically and by orientation-preserving isometries on (mathcal {A},mathcal {H},D) and a unitary fiber functor psi on {mathbb {G}}. The deformation procedure is a proper generalization of the cocycle deformation of Goswami and Joardar.

Twisting Hopf algebras from cocycle deformations

Let H be a Hopf algebra. Any finite-dimensional lifting of $$V\in {}^{H}_{H}\mathcal {YD}$$ arising as a cocycle deformation of $$A={\mathfrak {B}}(V)\#H$$ defines a twist in the Hopf algebra $$A^*$$ , via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.

Deformation by cocycles of pointed Hopf algebras over non-abelian groups

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.

Finite-dimensional pointed or copointed Hopf algebras over affine racks

We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification for the six affine racks whose Nichols algebra is known to be of finite dimension. In the pointed case our method allows us to finish four of them. In all of the cases the Hopf algebras obtained turn out to be cocycle deformations of their associated graded Hopf algebras. All of them are new examples of finite-dimensional copointed or pointed Hopf algebras over non-abelian groups.

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.

On a family of Hopf algebras of dimension 72

We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on S_3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.

Cocycle Deformations for Liftings of Quantum Linear Spaces

Let A be a Hopf algebra over a field K of characteristic 0, and suppose there is a coalgebra projection π from A to a sub-Hopf algebra H that splits the inclusion. If the projection is H-bilinear, then A is isomorphic to a biproduct R#ξ H where (R, ξ) is called a pre-bialgebra with cocycle in the category . The cocycle ξ maps R ⊗ R to H. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points Γ as classified by Andruskiewitsch and Schneider [1]. One asks when such an A can be twisted by a cocycle γ: A ⊗ A → K to obtain a Radford biproduct. By results of Masuoka [12, 13], and Grünenfelder and Mastnak [10], this can always be done for the pointed liftings mentioned above. In a previous article [3], we showed that a natural candidate for a twisting cocycle is λ ○ ξ where λ ∈H* is a total integral for H and ξ is as above. We also computed the twisting cocycle explicitly for liftings of a quantum linear plane and found some examples where the twisting cocycle we computed was different from λ ○ ξ. In this note, we show that in many cases this cocycle is exactly λ ○ ξ and give some further examples where this is not the case. Also, we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension.