Equivariant Floer groups for binary polyhedral spaces

This work was partially supported by a grant from the National Sciences and Engineering Research Council     

Normal Hopf subalgebras in cocycle deformations of finite groups

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = H ^σ of the Hopf algebra H = k ^G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra corresponds to a Hopf subalgebra . Our main result is an explicit necessary and sufficient condition for the normality of L′ in L. This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).     

Hopf algebra forms of the multiplicative group and other groups

The multiplicative group functor, which associates with each k-algebra its group of units, is affine with Hopf algebra k[x,x^–1]. The purpose of this paper is to determine explicitly all Hopf algebra forms of k[x,x^–1] with only minor restrictions on k (2 not a zero-divisor and Pic₍₂₎(k)=0). We also describe explicitly (by generators and relations) the Hopf algebra forms of kC₃, kC₄ and kC₆, where C_n is the cyclic group of order n. Some of our results could be drawn from [1,III §5.3.3] where a similar result as ours is indicated (and left as an exercise). We prefer however a less technical approach, in particular we do not use the extended theory of algebraic groups and functor sheaves.     

Hopf algebra actions

Finite groups acting on rings by automorphisms, and group-graded rings are instances of Hopf algebras H acting on H-module algebras A. We study such actions. Let A^H = {a ε A¦h · A = (h) a, all h ε H}, the ring of H-invariants, and form the smash product A # H. We study the ring extensions A^H A A # H. We prove a Maschke-type theorem for A # H-modules. We form an associated Morita context [A^H, A, A, A # H] and use these to get connections between the various rings.     

The Hopf algebra of diagonal rectangulations

We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a Hopf subalgebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

On the Hopf algebra of multi-complexes

Journal of Algebraic Combinatorics - We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and...     

The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

Twisting Hopf algebras from cocycle deformations

Let H be a Hopf algebra. Any finite-dimensional lifting of \(V\in {}^{H}_{H}\mathcal {YD}\) arising as a cocycle deformation of \(A={\mathfrak {B}}(V)\#H\) defines a twist in the Hopf algebra \(A^*\), via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

Cotorsion dimensions and Hopf algebra actions

Let H be a finite-dimensional Hopf algebra over a field k, and let A be an H-module algebra. In this paper, we discuss the cotorsion dimension of the smash product A # H. We prove that $$l.\cot .D\left( {A\# H} \right) \leqslant l.\cot .D\left( A \right) + r.D\left( H \right),$$ which generalizes the result of group rings. Moreover, we give some sufficient conditions for which $$l.\cot .D\left( {A\# H} \right) = l.\cot .D\left( A \right).$$ As applications, we study the invariants of IF properties and Gorenstein global dimensions.     

A Hopf algebra for counting cycles

Cycles, also known as self-avoiding polygons, elementary circuits or simple cycles, are closed walks which are not allowed to visit any vertex more than once. We present an exact formula for enumerating such cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from a Hopf algebra, which we construct explicitly, and which provides further means of counting cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate cycles.     

Quantum field theory meets Hopf algebra

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S ‐matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and leads to a correspondence between Feynman diagrams and semi‐standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S (V) to V. In many cases, noncommutative analogues are derived (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)