arXiv subjects
Publications and source records attributed to Kurusch Ebrahimi-Fard.
The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. In this context the notion of Rota-Baxter algebras enters the scene. We review several aspects of Rota-Baxter algebras as they appear in other sectors also relevant to perturbative renormalization, for instance multiple-zeta-values and matrix differential equations.
We establish Sakakibara's differential equations in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.
We study multiple zeta values and their generalizations from the point of view of Rota--Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota--Baxter algebras.
The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.
Coherent unit actions on a binary, quadratic operad were introduced by Loday and were shown by him to give Hopf algebra structures on the free algebras when the operad is also regular with a splitting of associativity. Working with such operads, we characterize coherent unit actions in terms of linear equations of the generators of the operads. We then use these equations to give all possible operad relations that allow such coherent unit actions. We further show that coherent unit actions are preserved under taking products and thus yield Hopf algebras on the free object of the product operads when the factor operads have coherent unit actions. On the other hand, coherent unit actions are never preserved under taking the dual in the operadic sense except for the operad of associative algebras.
The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.
Since its introduction by Loday in 1995 with motivation from algebraic K-theory, dendriform dialgebras have been studied quite extensively with connections to several areas in mathematics and physics. A few more similar structures have been found recently, such as the tri-, quadri-, ennea- and octo-algebras, with increasing complexity in their constructions and properties. We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures from known structures and from linear operators.
In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.
We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.
In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e. the full Hopf algebra of rooted trees or Feynman graphs is briefly outlined.
In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product.
In this brief note we would like to report on an observation concerning the relation between Rota-Baxter operators and Loday-type algebras, i.e. dendriform di- and trialgebras. It is shown that associative algebras equipped with a Rota-Baxter operator of arbitrary weight always give such dendriform structures.
In this note we comment on part of a recent article by B. Schroer and H.-W. Wiesbrock. Therein they calculate some new modular structure for the U(1)-current-algebra (Weyl-algebra). We point out that their findings are true in a more general setting. The split-property allows an extension to doubly-localized algebras.