Recent elegant work[1] on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relat...

Let H be a finite-dimensional quasi-Hopf algebra. We show for each quotient quasibialgebra Q of H that Q is a quasi-Hopf algebra whose dimension divides the dimension of H.

We discuss a method to construct a De Rham complex (diierential algebra) of Poincar e-Birkhoo-Witt-type on the universal enveloping algebra of a Lie algebra g. We determine the cases in which this gives rise to a diierential Hopf algebra that naturally extends the Hopf algebra structure of U(g). The construction of such diierential structures is interpreted in terms of colour Lie superalgebras.

In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk Kreimer in [3]; the Hopf algebra which occurs, denoted by HR, is the polynomial algebra in an infin...

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H 1 , the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or...

We define the Hopf algebra structure on the Grothendieck group of finitedimensional polynomial representations of Uq ĝlN in the limit N → ∞. The resulting Hopf algebra Rep Uq ĝl∞ is a tensor product of its Hopf subalgebras Repa Uqĝl∞, a ∈ C/q. When q is generic (resp., q is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra Repa Uq ĝl∞ and the algebra of...

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corre...

A Hopf algebra object in Loday and Pirashvili’s category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel’d module. We equip the latter with a structure of a braided Leibniz algebra. This provides a uni ed framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator x is invertible and furthermore working polynomials in ln x instead of polynomials in x. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra.