Finite-dimensional Semisimple Hopf Algebras Research Articles

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.

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.

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.

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.

The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras

AbstractWe study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

On matched pairs defining Hopf algebras in a certain class

In previous work by the author, a class of finite-dimensional semisimple Hopf algebras was considered with respect to the question under what condition all but one isomorphism class of simple modules are one-dimensional. The group theoretical answer given there asks for a classification of certain matched pairs used for the construction of these Hopf algebras. This classification is the content of this paper.

Actions of Ore extensions and growth of polynomial H-identities

ABSTRACTWe show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexpH(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.

Kitaev Lattice Models as a Hopf Algebra Gauge Theory

We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models.

Biproducts and Kashina's Examples

We revisit a class of examples described in the original article on biproducts, expand the class, and provide a detailed analysis of the coalgebra and algebra structures of many of these examples. Connections with the semisimple Hopf algebras whose dimension is a power of two determined by Kashina are examined. The finite-dimensional nontrivial semisimple cosemisimple Hopf algebras we construct are shown to be lower cosolvable. Some of these have one proper normal Hopf subalgebra and are not lower solvable.

On a Class of Finite-dimensional Semisimple Hopf Algebras

Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, [1] and [9]. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.

THE CALABI–YAU PROPERTY OF TWISTED SMASH PRODUCTS

Let H be a finite-dimensional cocommutative semisimple Hopf algebra and A * H a twisted smash product. The Calabi–Yau (CY) property of twisted smash product is discussed. It is shown that if A is a CY algebra of dimension dA, a necessary and sufficient condition for A * H to be a CY Hopf algebra is given.

Amitsur’s conjecture for associative algebras with a generalized Hopf action

We prove the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras over a field of characteristic 0, including G-identities for any finite (not necessarily Abelian) group G and H-identities for a finite dimensional semisimple Hopf algebra H. In addition, we prove that the Hopf PI-exponent of Sweedler’s 4-dimensional algebra with the action of its dual equals 4.

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1–13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.

On finite-dimensional semisimple Hopf algebras of dimension n(n+ 1)

We study finite-dimensional semisimple Hopf algebras over an algebraically closed field which have only one summand of dimension greater than 1 in their semisimple decompositions and assume that the group of group elements in the dual Hopf algebra is cyclic and has minimal order. Under given constraints, we obtain a detailed description of the comultiplication and the antipode.

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.

Some Computations of Frobenius–Schur Indicators of the Regular Representations of Hopf Algebras

We study Frobenius–Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational results, we formulate the theorem of Frobenius for semisimple Hopf algebras and give some partial results on this problem.

DERIVED H-MODULE ENDOMORPHISM RINGS

AbstractLet H be a Hopf algebra, A/B be an H-Galois extension. Let D(A) and D(B) be the derived categories of right A-modules and of right B-modules, respectively. An object M⋅ ∈ D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings EA(M⋅) = ⊕i∈ℤHomD(A)(M⋅, M⋅[i]) and EB(M⋅) = ⊕i∈ℤHomD(B)(M⋅, M⋅[i]). If H is a finite-dimensional semi-simple Hopf algebra, then EA(M⋅) is a graded sub-algebra of EB(M⋅). In particular, if M is a usual A-module, a necessary and sufficient condition for EB(M) to be an H*-Galois graded extension of EA(M) is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.

Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field k by using the braiding structures of A . The coefficients of polynomial invariants are integers if k is a finite Galois extension of Q , and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q . Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A , and recognize the difference between the representation category and the representation ring of A . Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.

Gorenstein subrings of invariants under Hopf algebra actions

This paper concerns conditions on the action of a finite dimensional semisimple Hopf algebra on an Artin–Schelter regular algebra that force the subring of invariants to satisfy the Artin–Schelter Gorenstein condition. Classical results such as Watanabe's Theorem and Stanley's Theorem are extended from the case of a group action to the context of a Hopf algebra action. A Hopf algebra version of the homological determinant is introduced, and it becomes an important tool in the generalization from group actions to Hopf algebra actions.

The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H* are semisimple, then gl.dim(A#σH)=gl.dim(A), where σ is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim(C ⋊αH)=gl.dim(C).