3,500 research outputs found
Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications
We consider Rota-Baxter algebras of meromorphic forms with poles along a
(singular) hypersurface in a smooth projective variety and the associated
Birkhoff factorization for algebra homomorphisms from a commutative Hopf
algebra. In the case of a normal crossings divisor, the Rota-Baxter structure
simplifies considerably and the factorization becomes a simple pole
subtraction. We apply this formalism to the unrenormalized momentum space
Feynman amplitudes, viewed as (divergent) integrals in the complement of the
determinant hypersurface. We lift the integral to the Kausz compactification of
the general linear group, whose boundary divisor is normal crossings. We show
that the Kausz compactification is a Tate motive and that the boundary divisor
and the divisor that contains the boundary of the chain of integration are
mixed Tate configurations. The regularization of the integrals that we obtain
differs from the usual renormalization of physical Feynman amplitudes, and in
particular it may give mixed Tate periods in some cases that have non-mixed
Tate contributions when computed with other renormalization methods.Comment: 35 pages, LaTe
Pre-alternative algebras and pre-alternative bialgebras
We introduce a notion of pre-alternative algebra which may be seen as an
alternative algebra whose product can be decomposed into two pieces which are
compatible in a certain way. It is also the "alternative" analogue of a
dendriform dialgebra or a pre-Lie algebra. The left and right multiplication
operators of a pre-alternative algebra give a bimodule structure of the
associated alternative algebra. There exists a (coboundary) bialgebra theory
for pre-alternative algebras, namely, pre-alternative bialgebras, which
exhibits all the familiar properties of the famous Lie bialgebra theory. In
particular, a pre-alternative bialgebra is equivalent to a phase space of an
alternative algebra and our study leads to what we called -equations in a
pre-alternative algebra, which are analogues of the classical Yang-Baxter
equation.Comment: 34 page
O-operators on associative algebras and associative Yang–Baxter equations
An O-operator on an associative algebra is a generalization of a Rota–Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an O-operator on a Lie algebra in the study of the classical Yang–Baxter equation. We introduce the concept of an extended O-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras. We study algebraic structures coming from extended O-operators. Continuing the work of Aguiar deriving Rota–Baxter operators from the associative Yang–Baxter equation, we show that its solutions correspond to extended O-operators through a duality. We also establish a relationship of extended O-operators with the generalized associative Yang–Baxter equation
- …