Finite-dimensional Hopf Algebras Research Articles

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. We describe all annihilator ideals of indecomposable [Formula: see text]-modules by generators. In particular, we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.

On Hopf algebras over basic Hopf algebras of dimension 24

We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension $24$ and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.

On non-counital Frobenius algebras

A Frobenius algebra is a finite-dimensional algebra [Formula: see text] which comes equipped with a coassociative, counital comultiplication map [Formula: see text] that is an [Formula: see text]-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map [Formula: see text] as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

On unimodular module categories

Let C be a finite tensor category and M an exact left C-module category. We call M unimodular if the finite multitensor category RexC(M) of right exact C-module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M is an endofunctor S on M together with a natural isomorphism Hom_(M,N)⁎≅Hom_(N,S(M)) for M,N∈M, where Hom_ is the internal Hom functor of M. In this paper, we discuss the case where C and M are the category of finite-dimensional left modules over a finite-dimensional Hopf algebra H and a finite-dimensional left H-comodule algebra L, respectively. We give an explicit description of the relative Serre functor of M and its twisted module structure in terms of the Frobenius structure of L. We also study pivotal structures on M and give some concrete examples.

Modified toric code models with flux attachment from Hopf algebra gauge theory

Kitaev’s toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be extended with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to extensions of the toric code. Here we consider the simple case where the gauge group is unchanged but furnished with a non-trivial quasitriangular structure (R-matrix), which modifies the construction of the gauge theory. This leads to some interesting phenomena; for example, the space of functions on the group becomes a non-commutative algebra. We also obtain simple Hamiltonian models generalizing the toric code, which are of the same overall topological type as the toric code, except that the various species of particles created by string operators in the model are permuted in a way that depends on the R-matrix. In the case of gauge theory, we find that the introduction of a non-trivial R-matrix amounts to flux attachment.

Davydov-Yetter cohomology, comonads and Ocneanu rigidity

Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad arising from the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity. Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra.

Hopf Quasigroup Galois Extensions and a Morita Equivalence

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product A#H is linked to the algebra of H invariants AH by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for A/AH to be Galois parallel in the case of H finite dimensional Hopf algebra.

HOPF CO-BRACE, BRAID EQUATION AND BICROSSED

In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C(A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel’d doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A ▷◁ H to be a Hopf co-brace if A or H is a Hopf co-brace

4–Manifold invariants from Hopf algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

Classification of finite-dimensional Hopf algebras over dual Radford algebras

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.

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

Some isomorphism results for graded twistings of function algebras on finite groups

We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally , we also consider the isomorphism problem for finite-dimensional Hopf algebras fitting into abelian cocentral extensions. We apply our classification results to a number of concrete examples involving special linear groups over finite fields, alternating groups and dihedral groups.

Finite-Dimensional Hopf Algebras

This paper mainly discussed various characterizations for the finite-dimensional Hopf algebras over algebraically closed field and has characteristic 0. And further showed that the order of antipode of the Hopf algebras is finite, but also provides a hint on how to estimate the order of the antipodes.

Invariants from the Sweedler power maps on integrals

For a finite-dimensional Hopf algebra A with a nonzero left integral Λ, we investigate a relationship between Pn(Λ) and PnJ(Λ), where Pn and PnJ are respectively the n-th Sweedler power maps of A and the twisted Hopf algebra AJ. We use this relation to give several invariants of the representation category Rep(A) considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep(K8), Rep(kQ8) and Rep(kD4), although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals λ in A⁎ and λJ in (AJ)⁎. This can be used to give a uniform proof of the remarkable result which says that the n-th indicator νn(A) is a gauge invariant of A for any n∈Z. We also use the expression for λJ to give an alternative proof of the known result that the Killing form of the Hopf algebra A is invariant under twisting. As a result, the dimension of the Killing radical of A is a gauge invariant of A.

Quantum double aspects of surface code models

We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double D( G) symmetry, where G is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of D( G) and combine this with D( G)-bimodule properties of open ribbon excitation spaces [Formula: see text] to show how open ribbons can be used to teleport information between their endpoints s0, s1. We give a self-contained account that builds on earlier work but emphasizes applications to quantum computing as surface code theory, including gates on D( S3). We show how the theory reduces to a simpler theory for toric codes in the case of [Formula: see text], including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalize to D( H) models based on a finite-dimensional Hopf algebra H, including site actions of D( H) and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.

The finite dual of commutative-by-finite Hopf algebras

AbstractThe finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra H is studied. Such a Hopf algebra H is an extension of an affine commutative Hopf algebra A by a finite dimensional Hopf algebra $\overline{H}$ . The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $\overline{H}^{\ast}$ by the finite dual $A^{\circ}$ of A. This decomposition is then further analysed using the Cartier–Gabriel–Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$ . The detailed consequences for a number of families of examples are then studied.

On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, qexp ( H ) = exp ⁡ ( H 0 ) ; 2) qexp ( H ⋊ k 〈 S 2 〉 ) = qexp ( H ) .

Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS

AbstractWe introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.