Bialgebra cohomology, pointed Hopf algebras, and deformations

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts.     

Universal constructions for Hopf algebras

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are the various subcategory-embeddings arsing naturally in this context reflexive or coreflexive? (c) When does a category of Hopf monoids have all limits or colimits? These problems are also shown to be intimately related. Particular emphasis is given to the case of Hopf algebras, i.e., when the chosen symmetric monoidal category is the category of modules over a commutative unital ring.     

Triangular braidings and pointed Hopf algebras

We consider an interesting class of braidings defined in [S. Ufer, PBW bases for a class of braided Hopf algebras, J. Algebra 280 (2004) 84-119] by a combinatorial property. We show that it consists exactly of those braidings that come from certain Yetter-Drinfeld module structures over pointed Hopf algebras with abelian coradical.As a tool we define a reduced version of the FRT construction. For braidings induced by U_q(g)-modules the reduced FRT construction is calculated explicitly.     

The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.     

Finitely generated invariants of Hopf algebras on free associative algebras

We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.     

Cyclic homology of Hopf crossed products

We obtain a mixed complex, simpler than the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one works in the general setting and the second one (which generalizes those previously found by several authors) works when f takes its values in K.     

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.     

Rank varieties for a class of finite-dimensional local algebras

We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, . Included in this class are the truncated polynomial algebras , with k an algebraically closed field and arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander-Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context.     

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     

Hopf algebras constructed by the FRT-construction

Given any bialgebra A and a braiding product 〈|〉 on A, a bialgebra U_〈|〉 was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”, Comm. Algebra 19 (1991) 3295-3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97-110]. In the present paper it is proved that when U_〈|〉 is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra.     

Decomposition of some pointed Hopf algebras given by the canonical Nakayama automorphism

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.     

Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.     

Co-Frobenius Hopf algebras and the coradical filtration

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. Let H be a Hopf algebra whose coradical is a Hopf algebra. Let gr H be the associated graded coalgebra and let R be the diagram of H, c. f. [2]. Then the following are equivalent: (1) H is co-Frobenius; (2) gr H is co-Frobenius; (3) R is finite dimensional; (4) the coradical filtration of H is finite. This Theorem allows us to give systematically examples of co-Frobenius Hopf algebras, and opens the way to the classification of ample classes of such Hopf algebras. Received: 8 March 2001 / Published online: 8 November 2002 RID="*" ID="*" Parts of this work were done during visits of the second author to the University of Córdoba, Argentina, in May 2000; and of the first author to the University of Bucharest in November 2000. We thank the FOMEC and CNCSIS (Grant C12) for support to these visits. The first author also thanks ANPCyT, CONICET, Agencia Córdoba Ciencia and Secyt (UNC) for partial support. The second author was partiall y supported by Grant SM10/01 of the Research Administration of Kuwait University.     

Almost-triangular Hopf Algebras

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra with (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field . We denote by the vector space generated by the left tensorand of . Then is a sub-Hopf algebra of . We proved that when is odd, has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when is even, is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.      

Twisted support varieties

We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis we show that the twisted variety of a module satisfies Dade’s Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.     

Hochschild cohomology of triangular string algebras and its ring structure

We compute the Hochschild cohomology groups HH^?(A)${\mathrm{HH}}^{?}(A)$ in case A is a triangular string algebra, and show that its ring structure is trivial.     

From racks to pointed Hopf algebras

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces , where X is a rack and q is a 2-cocycle on X with values in . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.     

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.     

Mould calculus,polyhedral cones,and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation.