Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

The Brauer-Clifford group for locally finite (S,H)-Azumaya algebras

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.     

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.     

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.     

The Hopf Algebra of Rooted Trees in Epstein-Glaser Renormalization

We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator B₊.submitted 29/03/04, accepted 01/06/04     

The wedderburn-malcev theorem for comodule algebras

Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.     

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors ? ? _k V and V ? _k  ? are known to be left adjoint to some kind of Hom-functors as endofunctors of _H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case.

In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors ? ?V and ? ?V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors ? ?V* and Hom _k (V, ?) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.     

Algebraic density property of Danilov–Gizatullin surfaces

A Danilov–Gizatullin surface is an affine surface V which is the complement of an ample section S for the ruling of a Hirzebruch surface. The remarkable theorem of Danilov and Gizatullin states that the isomorphism class of V depends only on the self-intersection number (S.S). In this paper we apply the theorem of Danilov–Gizatullin to prove that the Lie algebra generated by the complete algebraic vector fields on V coincides with the set of all algebraic vector fields of V.     

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of ( F , G )-dimodules associated to two functors between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.      

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.     

Fractional supersymmetric su(2) algebras

In this paper, the third root of the Lie algebra su(2) based on the permutation group S₃ is formulated in the Hopf algebra formalism.     

Hopf algebras with rigid dualizing complexes

Let H be a Hopf algebra over a base field. If H has an ℕ-filtration such that the associated graded ring is connected graded noetherian and has enough normal elements, then H is Gorenstein. This gives a partial solution to a question of Brown and Brown-Goodearl. As a consequence, every quotient Hopf algebra of a generic quantized coordinate ring of a connected semisimple Lie group is Auslander-Gorenstein and Cohen-Macaulay. The last statement answers a question of Goodearl-Zhang.     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

A Frobenius–Schur Theorem for Hopf Algebras

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.     

GK dimension of some connected Hopf algebras

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.     

Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations. I thank my family and FTHD organizer S.L. Eriksson. Soli Deo Gloria