Showing papers on "Hopf algebra published in 1998"

Hopf Algebras, Renormalization and Noncommutative Geometry

Abstract: We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem

Abstract: In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific Hopf algebra \(\), associated to each integer codimension. This Hopf algebra reduces transverse geometry, to a universal geometry of affine nature. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper, in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the Hochschild cohomology of a Hopf algebra carries a highly nontrivial cyclic structure.

On the Hopf algebra structure of perturbative quantum field theories

Abstract: We show that the process of renormalization encapsules a Hopf algebra structure in a natural manner. This sheds light on the recently proposed connection between knots and renormalization theory.

Quantum groups and quantum shuffles

Abstract: 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

Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3

Abstract: We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grad(A). Then grad(A) is a graded Hopf algebra, since the coradical A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A) \to A_0; let R be the algebra of coinvariants of \pi. Then, by a result of Radford and Majid, R is a braided Hopf algebra and grad(A) is the bosonization (or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle we propose to study A is first to study R, then to transfer the information to grad(A) via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p^3 (p an odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p^2; and an infinite family of pointed, non-isomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky.

Some properties of finite-dimensional semisimple Hopf algebras

Abstract: Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.

Hopf algebras, cyclic cohomology and the transverse index theory

Abstract: We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology.

Discrete spectral triples and their symmetries

Abstract: We classify zero-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf algebra structure of the finite algebra. We discuss examples of commutative algebras and group algebras.

Braided Hopf algebras over non abelian finite groups

Abstract: This is a survey of general aspects of the theory of braided Hopf algebras with emphasis on a special class of braided graded Hopf algebras named tobas. The interest on tobas arises from problems of classification of pointed Hopf algebras. We discuss tobas from different points of view following ideas of Lusztig, Nichols and Schauenburg. We then concentrate on braided Hopf algebras in the Yetter-Drinfeld category over H, where H is the group algebra of a non abelian finite group. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider.

Quantum Double for Quasi-Hopf Algebras

Abstract: We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double Dφ(G) associated to a finite group G and group 3-cocycle φ.

On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic

Abstract: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky's 5th conjecture on the order of the antipode of A. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish.

Problems in the steenrod algebra

Abstract: This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 452 2 Differential operators and integral Steenrod squares 460 3 Symmetric functions and differential operators 471 4 Bases, excess and conjugates 476 5 The stripping technique 483 6 Iteration theory and nilpotence 490 7 The hit problem and invariant theory 495 8 The dual of A(n) and graph theory 504 9 The Steenrod group 507 10 Computing in the Steenrod algebra 508 References 511 In Section 1 the scene is set with a few remarks on the early history of the Steenrod algebra A at the prime 2 from a topologist's point of view, which puts into context some of the problems posed later. In Section 2 the subject is recast in an algebraic framework, by citing recent work on integral versions of the Steenrod algebra defined in terms of differential operators. In Section 3 there is an explanation of how the divided differential operator algebra D relates to the classical theory of symmetric functions. In Section 4 some comments are made on a few of the recently discovered bases for the Steenrod algebra. The stripping technique in Section 5 refers to a standard action of a Hopf algebra on its dual, which is particularly useful in the case of the Steenrod algebra for deriving relations from relations when implemented on suitable bases. In Section 6 a parallel is drawn between certain elementary aspects of the iteration theory of quadratic polynomials and problems about the nilpotence height of families of elements in the Steenrod algebra. The hit problem in Section 7 refers to the general question in algebra of finding necessary and sufficient conditions for an element in a graded module over a graded ring to be decomposable into elements of lower grading. Equivariant versions of this problem with respect to general linear groups over finite fields have attracted attention in the case of the Steenrod algebra acting on polynomials. Similar problems arise with respect to the symmetric groups and the algebra D. This subject relates to topics in classical invariant theory and modular representation theory. In Section 8 a number of statements about the dual Steenrod algebra are transcribed into the language of graph theory. In Section 9 a standard method is employed for passing from a nilpotent algebra over a finite field of characteristic p to a p-group, and questions are raised about the locally finite 2-groups that arise in this way from the Steenrod algebra. Finally, in Section 10 there are a few comments on the use of a computer in evaluating expressions and testing relations in the Steenrod algebra. 1991 Mathematics Subject Classification 55S10.

Axioms for Weak Bialgebras

Abstract: Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is multiplicative. We do not require \Delta(1) = 1 \otimes 1 nor multiplicativity of the counit \eps. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that \Rep\A becomes a monoidal category with unit object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat A denoting the dual weak bialgebra). Under these monoidality axioms \E and \bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are trivial if and only if the counit \eps is multiplicative. We also propose axioms for an antipode S such that the category \Rep\A becomes rigid. S is uniquely determined, provided it exists. If a monoidal weak bialgebra A has an antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra anti-homomorphism, in which case S is also invertible. In this way we obtain a definition of weak Hopf algebras which in Appendix A will be shown to be equivalent to the one given independently by G. B\"ohm and K. Szlach\'anyi. Special examples are given by the face algebras of T. Hayashi and the generalised Kac algebras of T. Yamanouchi.

K 0-rings and twisting of finite dimensional semisimple hopf algebras

Abstract: We show that the K 0-group of a finite dimensional semisimple Hopf algebra has a natural structure of a ring with involution and prove that this ring is a twisting invariant of the Hopf algebra.We also study relations between algebraic structure of a Hopf algebra and the one of its K 0ring and prove that the twisting of a finite simple group is a simple Hopf algebra (i.e., it does not have proper normal Hopf subalgebras).In this case the dual Hopf algebra does not have proper Hopf subalgebras.Further, we construct several series of non trivial Hopf algebras by the twisting of the classical series of finite groups and give examples of Hopf algebras which can not be described as twisting of any group.

Brauer Groups, Hopf Algebras and Galois Theory

Abstract: I: The Brauer Group of a Commutative Ring. 1. Morita Theory for Algebras without a Unit. 2. Azumaya Algebras and Taylor-Azumaya Algebras. 3. The Brauer Group. 4. Central Separable Algebras. 5. Amitsur Cohomology and etale Cohomology. 6. Cohomological Interpretation of the Brauer Group. II: Hopf Algebras and Galois Theory. 7. Hopf Algebras. 8. Galois Objects. 9. Cohomology over Hopf Algebras. 10. The Group of Galois (co)Objects. 11. Some Examples. III: The Brauer-Long Group of a Commutative Ring. 12. H-Azumaya Algebras. 13. The Brauer-Long Group of a Commutative Ring. 14. The Brauer Group of Yetter- Drinfel'd Module Algebras. A: Abelian Categories and Homological Algebra. B: Faithfully Flat Descent. C: Elementary Algebraic K-Theory.

Actions of Multiplier Hopf Algebras

Abstract: For an action $\alpha$ of a group $G$ on an algebra $R$ (over $\Bbb C$), the crossed product $R\times_\alpha G$ is the vector space of $R$-valued functions with finite support in $G$, together with the twisted convolution product given by $$(\xi \eta)(p) = \sum_{q \in G} \xi(q) \alpha_q (\eta (q^{-1}p))$$ where $p\in G$. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra $A$ on an algebra $R$, it is possible to make the tensor product $R\ot A$ into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by $R# A$. In the group case, the action $\alpha$ of $G$ on $R$ yields an action of the group algebra $\Bbb C G$ as a Hopf algebra on $R$ and the crossed $R\times_\alpha G$ coincides with the smash product $R# \Bbb C G$. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf algebras to formulate this duality theorem. We prove a result in the case of an algebraic quantum group and its dual. The more general case is only stated and will be proven in a separate paper on coactions. These duality theorems for actions are substantial generalizations of the corresponding theorem for Hopf algebras. Also the techniques that are used here to prove this result are slightly different and simpler.

On Finite-Dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic

Abstract: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky's 5th conjecture on the order of the antipode of A. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish.

Elliptic Quantum Groups $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ and Quasi-Hopf Algebras

Abstract: We construct an algebra morphism from the elliptic quantum group $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ to a certain elliptic version of the “quantum loop groups in higher genus” studied by V. Rubtsov and the first author. This provides an embedding of $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ in an algebra “with central extension”. In particular we construct L±-operators obeying a dynamical version of the Reshetikhin–:Semenov-Tian-Shansky relations. To do that, we construct the factorization of a certain twist of the quantum loop algebra, that automatically satisfies the “twisted cocycle equation” of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang–Baxter equation.

Algebraic versions of a finite-dimensional quantum groupoid.

Abstract: We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T Yamanouchi, a weak $C^*$-Hopf algebra introduced by G Bohm, F Nill and K Szlachanyi (with an involutive antipode), and a Kac bimodule -- an algebraic version of a Hopf bimodule, the notion introduced by J-M Vallin We also study the structure and construct examples of finite dimensional quantum groupoids