Semisimple Hopf 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.

Equivariant Fusion Subcategories

AbstractWe provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

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.

Representation rings of extensions of Hopf algebra of Kac-Paljutkin type

<p>In this paper, we focus on studying two classes of finite dimensional $ \Delta $-associative algebras, which are extensions of a family of $ 2n^2 $-dimensional Kac-Paljutkin type semi-simple Hopf algebras $ H_{2n^2} $. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.</p>

Fermionization of fusion category symmetries in 1+1 dimensions

We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u), which is the supercategory of super representations of a weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u. The weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u depends not only on H but also on a choice of a non-anomalous ℤ2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u by explicitly constructing fermionic TQFTs with SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.

An Indicator Formula for the Hopf Algebra kSn-1#kCn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k^{S_{n-1}}\\#kC_n$$\\end{document}

The semisimple bismash product Hopf algebra Jn=kSn-1#kCn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_n=k^{S_{n-1}}\\#kC_n$$\\end{document} for an algebraically closed field k is constructed using the matched pair actions of Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_n$$\\end{document} and Sn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{n-1}$$\\end{document} on each other. In this work, we reinterpret these actions and use an understanding of the involutions of Sn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{n-1}$$\\end{document} to derive a new Froebnius-Schur indicator formula for irreps of Jn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_n$$\\end{document} and show that for n odd, all indicators of Jn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_n$$\\end{document} are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all 2-dimensional irreps of Jn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_n$$\\end{document} and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of Jn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_n$$\\end{document} and use these formulas to show that nonzero indicators become rare for large n.

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.

Gorenstein Flat Modules of Hopf-Galois Extensions

Let A/B be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications.

Hopf Differential Graded Galois Extensions

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that D(R#H)≅D(RH) is equivalent to that R/RH is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions.

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

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.

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.

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

On lattice models of gapped phases with fusion category symmetries

We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an H-simple left H-comodule algebra, where H is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries mathcal{C} on the boundary conditions of these state sum TQFTs are represented by module categories over mathcal{C} . This agrees with the classification of gapped phases with symmetry mathcal{C} . We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.

Degree bounds for Hopf actions on Artin–Schelter regular algebras

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

On PI-algebras with additional structures: Rationality of Hilbert series and Specht's problem

One of the main problems in PI-theory is to prove the rationality of the Hilbert series of the relatively free algebra of a given PI-algebra. In this paper we consider a field F of characteristic 0 and we prove the rationality of the Hilbert series of the PI-algebra A over F both in the case A is a superalgebra with superinvolution and when a finite dimensional semisimple Hopf algebra acts on A. Along the way, we give a proof of the Specht's problem in case A is a superalgebra with superinvolution.