Quiver Hopf algebras

In this paper we study subHopfalgebras of graded Hopf algebra structures on a path coalgebra kQ^c. We prove that a Hopf structure on some subHopfquivers can be lifted to a Hopf structure on the whole Hopf quiver. If Q is a Schurian Hopf quiver, then we classified all simple-pointed subHopfalgebras of a graded Hopf structure on kQ^c. We also prove a dual Gabriel theorem for pointed Hopf algebras.     

L'algebre de hopf bitensorielle

For any field k of zero characteristic we give a functor from the category of k-vector spaces into the category of k-Hopf algebras, attaching to any vector space V its bitensorial pointed Hopf algebra A^v. This Hopf algebra is graded, fulfills a universal property, and contains a remarkable subspace P of primitive elements, which as a conjecture may generate the Lie algebra Prim A^v. In case V is finite-dimensional we exhibit a Hopf pairing between A^vand A^v-whose kernel contains the (Hopf) ideal generated by the elements of P of degree ? 2.     

Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

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.     

Techniques for Classifying Hopf Algebras and Applications to Dimension p 3

Classifying Hopf algebras of a given finite dimension n over ? is a challenging problem. If n is p, p², 2p, or 2p² with p prime, the classification is complete. If n = p³, the semisimple and the pointed Hopf algebras are classified, and much progress on the remaining cases was made by the second author but the general classification is still open. Here we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p³ and finish the classification for dimension 27.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

Quantum groups and quantum shuffles

Let U _q ⁺ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show that U _q ⁺ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z ⁿ . This method gives supersymetric as well as multiparametric versions of U _q ⁺ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U _q sl(2) and a suitable irreducible finite dimensional representation. Oblatum 21-III-1997 & 12-IX-1997     

Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

We extend the Larson–Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A^T of the underlying weak Hopf algebra A.     

Finite dimensional Nichols algebras over certain p-groups

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.     

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.     

DIMENSION DEFECTS AND THE HOPF INVARIANT

Abstract

The concept of dimension defect of a mapping was introduced by H. Hopf in [5]. We generalize and answer questions about mappings S³ → S² which he raised at the end of that paper. Our main result is that a mapping S^2n-1 → Sⁿ with non-vanishing Hopf invariant does not have dimension defect.     

On the Structure of Weak Hopf Algebras

We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of Tr(S²) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.     

Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

On Semisimple Hopf Algebras of Dimension pq2, II

We obtain further classification results for semisimple Hopf algebras of dimension pq ² over an algebraically closed field k of characteristic zero. We complete the classification of semisimple Hopf algebras of dimension 28.     

On almost complex curves and Hopf hypersurfaces in the nearly Khler six-sphere

We study the almost complex curves and Hopf hypersurfaces in the nearly Khler S6(1),and their relations.For Hopf hypersurfaces,we give a classification theorem under some additional conditions.For compact almost complex curves,we obtain some interesting global results with respect to Gaussian curvature,area and the genus.     

Duality Features of Left Hopf Algebroids

We explore special features of the pair (U ^?,U _?) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S ^? from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phùng’s categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products