Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B⁺(P(X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees HNCK, where B⁺ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.     

The complexity of crossed products

Let H be a finite-dimensional Hopf algebra, let A be a finite-dimensional algebra measured by H, and let A # _σ H be a crossed product. In this paper, we first show that if H is semisimple as well as its dual H*, then the complexity of A # _σ H is equal to that of A. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra H is equal to the complexity of the trivial module _H k. As an application, we prove that the complexity of Sweedler’s 4-dimensional Hopf algebra H ₄ is equal to 1.     

Basic quasi-Hopf algebras over cyclic groups

Let m be a positive integer, not divisible by 2, 3, 5, 7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in Etingof and Gelaki (2006) [11] to the case of cyclic groups of order m. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras A(H,s), constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order m is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type A(H,s).     

Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H=kG: it is the least integer n for which the nth Hopf power endomorphism [n] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra H=O(G).The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina. Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki.Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group.In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Smash coproduct and braided groups in braided monoidal category

A smash coproduct in braided monoidal category C is constructed and some conditions making the smash coproduct a Hopf algebra or braided Hopf algebra are given. It is shown that the smash coproductB ×H in ^HM is equivalent to the transmutation of Hopf algebra. Thus a method for transmutation theory is provided. Let σ be 2-co-cycle andH a commutation Hopf algebra. A Hopf algebraHσ is constructed.Hσ?Hσ whereHσ is a transmutation ofHσ. The braided groups from some solutions of quantum Yang-Baxter equation are obtained.     

Transmutation theory of a coquasitriangular weak Hopf algebra

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.     

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.     

Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

Based on the ideas of Tannaka-Kre?n reconstruction, we present a categorical construction that assigns to any cleft Hopf algebra inclusion K⊂H a coquasibialgebra having K* as a Hopf subalgebra. As a special case, the construction gives an intrinsic connection between the bismash product K#Q and the double cross- product Q?K* constructed from the same combinatorial data. A cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a Sweedler three-cocycle. Thus our construction assigns to every bicrossproduct (or Hopf algebra extension) of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a Sweedler three-cocycle. Based on this observation we use the construction to prove generalizations of Kac's exact sequence for the group of Hopf algebra extensions of a group algebra by a dual group algebra.     

On twisted smash products for bimodule algebras and the drinfeld double 1

In this paper we construct a new algebra AHof an H- bimodule algebra Aby a Hopf algebra Hand study some of its properties. The smash product, the Drinfel'd double D(H) and the Doi-Takeuchi's algebra B?,H, are all special cases of AH. Moreover,we find a necessary and sufficient condition for A Hto be a Hopf algebra and also consider the dual situation     

Co-Poisson structures on polynomial Hopf algebras

The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.     

Actions of Ore extensions and growth of polynomial H-identities

We 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 PIexp^H(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.     

Extensions of the endomorphism algebra of weak comodule algebras

Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by using Hopf-Galois theory and Morita theory.     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

Pseudosymmetric braidings, twines and twisted algebras

A laycle is the categorical analogue of a lazy cocycle. Twines (introduced by Bruguières) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If c is a braiding, the double braiding c² is always a twine; we prove that it is a strong twine if and only if c satisfies a sort of modified braid relation (we call such cpseudosymmetric, as any symmetric braiding satisfies this relation). It is known that the category of Yetter-Drinfeld modules over a Hopf algebra H is symmetric if and only if H is trivial; we prove that the Yetter-Drinfeld category _HYD^H over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the 2ⁿ⁺¹-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by the properties of laycles and twines.     

Maschke Type Theorems for Nonassociative Hopf Algebras

In this article, we give necessary and sufficient conditions for a possibly nonassociative comodule algebra over a nonassociative Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the associative setting. Also, from this result we deduce a version of Maschke's Theorems and the consequent characterization of projectives for (H, B)-Hopf triples associated with a nonassociative Hopf algebra H and a nonassociative right H-comodule algebra B.     

Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z\({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)^H, where Z(A) is the center of algebra A, H⁰ is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)^H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A^H for a pointed Hopf algebra over a field of non-zero characteristic.     

Extensions of quantum enveloping algebras

In this paper, we define a class of extended quantum enveloping algebras U _q (f(K, J)) and some new Hopf algebras, which are certain extensions of quantum enveloping algebras by a Hopf algebra H. This construction generalizes some well-known extensions of quantum enveloping algebras by a Hopf algebra and provides a large of new noncommutative and noncocommutative Hopf algebras.     

Quantum doubles in monoidal categories

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ? _r H ^?cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and i_H H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.     

Homological dimension of weak Hopf�CGalois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of?A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of?B. As an application, we obtain Maschke-type theorems for weak Hopf?CGalois extensions and weak smash products.