A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions. 1. Introduction. It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed by Connes and Moscovici [5]...

We study the renormalization of massless QED from the point of view of the Hopf algebra discovered by D. Kreimer. For QED, we describe a Hopf algebra of renormalization which is neither commutative nor cocommutative. We obtain explicit renormalization formulas for the electron and photon propagators, for the vacuum polarization and the electron self-energy, which are equivalent to Zimmermann’s ...

By starting from the non-standard quantum deformation of the sl(2,R) algebra, a new quantum deformation for the real Lie algebra so(2, 2) is constructed by imposing the former to be a Hopf subalgebra of the latter. The quantum so(2, 2) algebra so obtained is realized as a quantum conformal algebra of the (1 + 1) Minkowskian spacetime. This Hopf algebra is shown to be the symmetry algebra of a t...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally assoc...

Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an a...

Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove...

William M. Singer has described a cohomology theory of connected Hopf algebras which classifies extensions of a cocommutative Hopf algebra by a commutative Hopf algebra in much the same way as the cohomology of groups classifies extensions of a group by an abelian group. We compute these cohomology groups for monogenic Hopf algebras, construct an action of the base ring on the cohomology groups...

Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ = Γ/J that contains Γ as its first order part. Corresponding to the three choices of the ideal J , we distinguish the ‘universal’ exterior algebra, the ‘second antisymmetrizer’ exterior algebra, and Woronowicz’ externa...

In a recent paper (1994 J. Phys. A: Math. Gen. 27 5907), Oh and Singh determined a Hopf structure for a generalized q-oscillator algebra. We prove that under some general assumptions, the latter is, apart from some algebras isomorphic to su q (2), su q (1,1), or their undeformed counterparts, the only generalized deformed oscillator algebra that supports a Hopf structure. We show in addition th...