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 ...

The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx 1. The Lie Algebra of Abelian Topological Groups Let G be an abelian topological group. The set of all one–parameter–subgroups L(G) := {λ : R→ G : λ is a continuous homomorphism} is called the Lie algebra of G . We define addition pointwise and scalar multi...

In a recent paper [1], a systematic construction of integrable Hamiltonians with coalgebra symmetry has been proposed. Such procedure can be applied to any Poisson coalgebra (A,∆) with generators Xi, i = 1, . . . , l and Casimir element C(X1, . . . , Xl) as follows. Let us consider the N -th coproduct ∆(Xi) of the generators and the m-th order (2 ≤ m ≤ N) coproducts ∆(m)(C) of the Casimir opera...

Let ? be a fixed complex number, and let simply connected region in plane C that is starlike with respect to ?. We define some Banach space of analytic functions on prove it algebra the ?-Duhamel product defined by (f?? g)(z) := d/dz z?? f(z+??t)g(t)dt. its maximal ideal consists homomorphism h? h?(f)=f(?). Further, we characterize lattice invariant subspaces integration operator J?f(z)=?z? f(t...

We show that the E-theory of Connes and Higson can be formulated in terms of C∗extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C∗-algebra A by a stable C∗algebra B asymptotically split if there exists an asymptoti...

Let X be a compact metric space and let α be a homeomorphism on X. Related to a theorem of Pimsner, we show that C(X) ⋊α Z can be embedded into a unital simple AF-algebra if and only if there is a strictly positive α-invariant Borel probability measure. Suppose that Λ is a Z action on X. If C(X)⋊Λ Z can be embedded into a unital simple AF-algebra, then there must exist a strictly positive Λ-inv...

In the semantics of programming, nite data types such as nite lists, have traditionally been modelled by initial algebras. Later nal coalgebras were used in order to deal with in nite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property ...

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...

In this paper we consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. By “deformations of a Lie algebra” we mean the (affine algebraic) manifold of all Lie brackets. Consider the quotient of this variety by the action of the group GL. It is well-known (see [Hart]) that in the category of algebraic varieties the quotient by a group action does no...

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category...