Search

that: "some of the me-thods used in Galois theory andformulated in the Hopf algebrasto Chase and Sweedler to defineterms of Hopf algebras whichdon 't formulate a corresponding theory of descent" . The pur-pose of this notes is to develop this theory, using asGalois definition of object, the one given by Chase andSweedler in

We study the realizations of certain braided vector spaces of rack type as Yetter–Drinfeld modules over a cosemisimple Hopf algebra H. We apply the strategy developed in [1] to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over S4. Fil: Garcia Iglesias, Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Vay, Cristian Damian. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space $V$ is naturally described by a linear operator $R$ on the algebra $\End V$ of linear transformations of~$V$. Double Lie algebras correspond in this sense to skew-symmetric Rota---Baxter operators, double associative algebra structures---to (left) averaging operators.

The Hopf dual $H^\circ$ 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.

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases., Comment: now covers the non-regular case; to appear in Comm. in Algebra

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebras. For the projectivity over Hopf subalgebras of a Hopf algebra we need some finiteness condition in terms of the module of integrals. In particular we show that the module of integrals is projective of rank one.

In this paper, for 𝒞 a strict braided monoidal category with tensor product ⊗, we improve the definition of Hom-associative algebra by removing the multiplicativity condition of the automorphism α....

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.

We introduce the theory IHR of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IHR are derived using Lack’s approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid–comonoid pairs. This construction is instru- mental in showing that IHR is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R

Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that both these convex monoids can be obtained from the other one by taking a coproduct of density matrices on the irreducible representations. We will also show that the same holds for a tensor product of a group and a function algebra., Comment: 13 pages

We continue the study begun by the third author of \(C^*\)-Segal algebra valued function algebras, with an emphasis on the order structure. Our main result is a characterization theorem for \(C^*\)-Segal algebra valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra valued function algebras. We also consider Gelfand representation of these algebras in the commutative case.

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k. We first construct a weak Hopf algebra [Δ(1)(H⊗H)Δ(1)]R, which is based on the subalgebra of the tensor product algebra H⊗H. Next we verify that if H is factorizable, then the Drinfeld’s quantum double of H is isomorphic to [Δ(1)(H⊗H)Δ(1)]R.

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H⁎ with respect to the Frobenius and symmetric properties.

We investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p3, p4 and p5, and list all the pointed Hopf algebras of order p6, whose group of grouplikes is non-abelian.

We study when a finite dimensional Hopf action on a quantum formal deformation A of a commutative domain A_0 (i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this occurs when the Poisson center of the fraction field of A_0 is trivial., Comment: v3. 9 pages. To appear in Proc. Amer. Math. Soc

Let H be a finite-dimensional quasi-Hopf algebra. We show for each quotient quasibialgebra Q of H that Q is a quasi-Hopf algebra whose dimension divides the dimension of H.

We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra QSym N ˜ of weak quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C. Rota relating symmetric type functions and Rota–Baxter algebras. We provide the transformation formulas between the weak monomial and fundamental quasi-symmetric functions, which extends the corresponding results for quasi-symmetric functions. Moreover, we show that QSym is a Hopf subalgebra and a Hopf quotient algebra of QSym N ˜ . Rota's question is addressed by identifying QSym N ˜ with the free commutative unitary Rota–Baxter algebra Ш ( x ) of weight 1 on one generator x, which also allows us to equip Ш ( x ) with a Hopf algebra structure.

Starting from any representation of the Lie algebra 𝔤 on the finite dimensional vector space V we can construct the representation on the space Aut(V). These representations are of the type of ad. That is one of the reasons, why it is important to study the adjoint representation of the Lie algebra 𝔤 on the universal enveloping algebra U(𝔤). A similar situation is for the quantum groups Uq(𝔤). In this paper, we study the adjoint representation for the simplest quantum algebra Uq(sl(2)) in the case that q is not a root of unity.

The purpose of this paper is to complete the classification of almost involutive Hopf algebras up to dimension 23. For this, we first present a brief survey of these algebras, showing that most of them can be obtained by iterated Hopf-Ore extensions. Then the classification follows using a technique we developed for determining when a Hopf-Ore extension is almost involutive.

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compare this invariant to the fundamental group of the underlying linear category, and we compute those groups for families of examples., Computations of the fundamental group of some Hopf algebras are added. The relation with the fundamental group of the underlying associative structure is now considered. We also analyse the situation when universal covers and/or gradings exist. Dedicated to Eduardo N. Marcos for his 60th birthday. 24 pages

In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.

This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully on a field. These Hopf algebras are referred to as Galois-theoretical. In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H of finite Cartan type. Namely, we determine when such H of type A[superscript ×r][subscript 1] and some H of rank two possess the Galois-theoretical property. Moreover, we provide necessary and sufficient conditions for Reshetikhin twists of small quantum groups to be Galois-theoretical. Keywords: Field, Finite Cartan type, Galois-theoretical, Hopf algebra action, Pointed, National Science Foundation (U.S.) (DMS-1502244), National Science Foundation (U.S.) (DMS-1550306)

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, H 0 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.

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\rightarrow 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., Comment: 24 pages, many figures

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. Bialgebra reducts of Hopf algebras are quantum loops, while sufficient conditions are given for quantum loop structure to augment to a Hopf algebra. The Moufang–Hopf algebras of Benkart et al., the Hopf quasigroups and coquasigroups of Klim–Majid, and the coassociative H -bialgebras of Perez-Izquierdo (for instance, the universal enveloping algebras of Sabinin algebras), all form quantum loops. Other quantum quasigroups offer natural nonassociative extensions of Hopf algebra constructions: quasigroup algebras, dual quasigroup algebras, and quantum couples of groups with quasigroups. Further examples include an algebra of rooted binary trees, and an algebra of skein polynomials.

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.

In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given., Comment: revised and updated

In this paper, we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism).

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements bik, cik, gik, hik (i ∈ I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra Uq(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A = (aij)i,j∈I. Therefore, we define an extended Hopf superalgebra H ⊗′ Uq(G). We also discuss the basis and the grouplike elements of H ⊗′ Uq(G).

In general, an intersection of two Hopf ideals of a Hopf algebra is not a Hopf ideal. By this reason, one may not define a Hopf ideal generated by a set of elements, and the Hopf algebras do not admit a usual combinatorial representation by generators and relations. Nevertheless, Heyneman–Radford theorem implies that each nonzero Hopf ideal of a pointed Hopf algebra has nonzero skew primitive element. Each ideal generated by skew primitive elements is a Hopf ideal. Therefore, the Heyneman–Radford theorem allows one to define a combinatorial representation step-by step by means of skew primitive relations. By definition the combinatorial rank is the minimal number of steps in that representation. We prove that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel of type D n , n > 3 equals ⌊ log 2 ⁡ ( 2 n − 3 ) ⌋ + 1 , provided that q has a finite multiplicative order greater than 2. It is already known that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel u q ( sl n + 1 ) of type A n equals ⌊ log 2 ⁡ n ⌋ + 1 , whereas in case u q ( so 2 n + 1 ) of type B n , it is equal to ⌊ log 2 ⁡ ( n − 1 ) ⌋ + 2 .

The aim of this paper is to study phase spaces of Hom-alternative algebras. We introduce notions of Hom-prealternative algebra and Hom-prealternative bialgebra. Bimodules and matched pairs of Hom-prealternative algebra are also considered. Furthermore, we show that the notion of phase space of a Hom-alternative algebra is equivalent to the notion of Hom-prealternative bialgebra. The coboundary Hom-prealternative bialgebra and Hom-PA equation are also described.

We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it.

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley's $(-1)$-color theorem. A $q$-analog version of this family of characters is also studied., Comment: To appear in SIAM J. of Discrete Math. This version has new material, including a proof of the antipode formula. The even and odd subalgebras section is corrected. An extended abstract version of this article appeared in the Proceedings of FPSAC'15

First, referring to our previous work, 'Hopf cyclic cohomology in braided monoidal categories', we reduce the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction, ψ 2 = id, on the braid map correspond to the Hopf algebra H, which is the main player in the theory. We define a family of examples of such desired braided Hopf algebras, H, living in the category of anyonic vector spaces. Next, on one hand, we will prove that these anyonic Hopf algebras are the enveloping (Hopf) algebras of particular quantum Lie algebras, which we will construct. On the other hand, we will show that, analogous to the non-super and the super case, the well known relationship between the periodic Hopf cyclic cohomology of an enveloping (super) algebra and the (super) Lie algebra homology also holds for these particular quantum Lie algebras.

We define the concept of companion automorphism of a Hopf algebra H as an automorphism \(\sigma :H\rightarrow H\) such that \(\sigma ^2=\mathcal {S}^2\), where \(\mathcal {S}\) denotes the antipode. This automorphism can be viewed as a special additional symmetry. A Hopf algebra is said to be almost involutive (AI) if it admits a companion automorphism. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non-almost involutive Hopf algebras.

We study generalised differential structures ( Ω 1 , d ) on an algebra A, where A ⊗ A → Ω 1 given by a ⊗ b → a d b need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs ( Λ 1 , ω ) where Λ 1 is a right module and ω a right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω : A + → Λ 1 in the category of right crossed (or Drinfeld–Radford–Yetter) modules over A. When A = U ( g ) the generalised left covariant differential structures are classified by cocycles ω ∈ Z 1 ( g , Λ 1 ) . We then introduce and study the dual notion of a codifferential structure ( Ω 1 , i ) on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra ( Ω , d ) augmented by a codifferential i of degree −1. Here Ω is a graded super-Hopf algebra extending the Hopf algebra Ω 0 = A and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. Accordingly, group 1-cocycles correspond precisely to codifferential structures on algebraic groups and function algebras. Among general constructions, we show that first order data ( Λ 1 , ω ) on a Hopf algebra A extends canonically to a strongly bicovariant differential graded algebra via the braided super-shuffle algebra. The theory is also applied to quantum groups with Ω 1 ( C q ( G ) ) dually paired to Ω 1 ( U q ( g ) ) .

The Jordan–Schwinger realization of quantum algebra U ⌣ q ( s u 2 ) is used to construct the irreducible submodule T l of the adjoint representation in two different bases. The two bases are known as types of irreducible tensor operators of rank l which are related to each other by the involution map. The bases of the submodules are equipped with q -analogues of the Hilbert–Schmidt inner product and it is also shown that the adjoint representation corresponding to one of those submodules is a *-representation.

An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with J. Ecalle's mould calculus, i.e. mould composition and contracting arborification, is precised., revised version, accepted for publication in Journal of Algebra

Comonoid, bi-algebra, and Hopf algebra structures are stud-ied within the universal-algebraic context of entropic varieties. Attentionfocuses on the behavior of setlike and primitive elements. It is shown thatentropic Jo´nsson-Tarski varieties provide a natural universal-algebraicsetting for primitive elements and group quantum couples (generaliza-tions of the group quantum double). Here, the set of primitive elementsof a Hopf algebra forms a Lie algebra, and the tensor algebra on anyalgebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then theunderlying Jo´nsson-Tarski monoid of the generating algebra is cancella-tive. The problem of determining when the Jo´nsson-Tarski monoid formsa group is open. 1. IntroductionThe aim of this paper is to consider comonoid, bi-algebra, and Hopf alge-bra structure within the context of universal algebra. The general categoricalsetting for such structure is provided by symmetric monoidal categories (or“symmetric tensor categories”, compare [11, p. 69]), with an associative tensorproduct and a unit object. Now an algebra (A,Ω) of type τ : Ω → Nis said tobe entropic if the operationω: A

The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of HopfR. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.

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.

A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed., Comment: LaTeX source, 39 pages

We study the Hopf *-algebra structures on the Hopf algebra H(1, q) over ℂ. It is shown that H(1, q) is a Hopf *-algebra if and only if |q| = 1 or q is a real number. Then the Hopf *-algebra structures on H(1, q) are classified up to the equivalence of Hopf *-algebra structures.

The paper discusses the structure of the Hopf cyclic homology and cohomology of the algebra of smooth functions on a manifold provided that the algebra is endowed with an action or a coaction of the algebra of Hopf functions on a finite or compact group or of theHopf algebra dual to it. In both cases, an analog of the Connes-Hochschild-Kostant-Rosenberg theorem describing the structure of Hopf cyclic cohomology in terms of equivariant cohomology and other more geometric cohomology groups is proved.

Let H be a finite-dimensional Hopf algebra acting on an algebra A. Then A is called an H-algebra whenever H acts on products in A via the coproduct. For example, when H=K[G] is a group Hopf algebra, then A is an H-algebra precisely when the group G acts by automorphisms on A; hence, if G acts by both automorphisms and anti-automorphisms on A, then A is not an H-algebra. In order to study such H-actions, we introduce the notion of an oriented H-algebra. An oriented Hopf algebra is a Hopf algebra with a specified $\mathbb {Z}_{2}$ -grading, H = H + ⊕ H −, while A is an oriented H-algebra whenever H + acts on products via the coproduct and H − acts on products via the ‘anti-coproduct’. This leads to an oriented convolution operation, ⋆. First, we characterize when ⋆ is associative. Then, when H is isomorphic to the dual of a group algebra of a finite abelian group, we provide certain duality theorems between oriented H-algebra actions on A and grading properties of the associated vector space decomposition of A. For example, a Lie algebra A is an oriented H-algebra if and only if A is quasigroup-graded with respect to ⋆. We show that there exists a quasigroup-grading of a Lie algebra that cannot be realized as a semigroup-grading, which was once thought impossible.

In this paper several classical facts known for group actions and group gradings on rings are extended to the case of a Noetherian H-module algebra A for a Hopf algebra H. When H is semisimple, a version of the Bergman-Isaacs result is proved, asserting the nilpotency of any onesided ideal of A whose intersection with the subalgebra of H-invariant elements A H is nilpotent. Under the additional assumption that A is H-semiprime, it is established that the classical quotient ring Q(A) of A is the Ore localization of A at the set of H-invariant regular elements. When H is finite-dimensional cosemisimple, the Jacobson radical of A is shown to be stable under the action of H. More generally, these results are obtained for algebras over an arbitrary commutative base ring under suitable restrictions on the Hopf algebra and its action.

We classify connected Hopf algebras of Gelfand-Kirillov dimension 4 over an algebraic closed field of characteristic zero.

We construct a functor $$T$$ from the category of endomorphisms of vector spaces to the category of Com-PreLie algebras. For any endomorphism $$f$$ of a vector space $$V$$ , we describe the enveloping algebra of the pre-Lie algebra $$T(V,f)$$ , the dual Hopf algebra and the associated group of characters. For $$f=\mathrm{Id}_V$$ , we find the algebra of formal diffeomorphisms, seen as a subalgebra of the Connes–Kreimer Hopf algebra of rooted trees in the context of QFT; for other well-chosen nilpotent $$f$$ , we obtain the groups of Fliess operators in Control Theory. An algebraic study of these Com-PreLie Hopf algebras is carried out: gradations, generation, subobject generated by $$V$$ , etc.

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras. As an application of a generalized Stone--von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions., 30 pages. v2: Minor changes. References added

The affine (Hopf) algebra of the special linear group SL(2l, k) for l ≥1 is not a free module over one of its Hopf subalgebras. In this paper we give a very simple proof of this fact over a field k of characteristic zero. The proof boils down to showing that a certain module is not cyclic.