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.
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples of Lie algebras to the context of Rota-Baxter Lie algebras. The coboundary case leads to the introduction of the admissible classical Yang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which the antisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of $\mathcal{O}$-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Lie …
Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the …
In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Li…
In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras pl…
In this paper, first we introduce the notion of a twisted Rota-Baxter operator on a 3-Lie algebra $\g$ with a representation on $V$. We show that a twisted Rota-Baxter operator induces a 3-Lie algebra structure on $V$, which represents on $\g$. By this fact, we define the cohomology of a twisted Rota-Baxter operator and study infinitesimal deformations of a twisted Rota-Baxter operator using the second cohomology group. Then we introduce the notion of an NS-3-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a twisted Rota-Baxter operator induces an NS-3…
From relative Rota-Baxter operators and relative averaging operators on Lie algebras to relative Rota-Baxter operators on Leibniz algebras: a uniform approach
We determine the \emph{$L_\infty$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes…
This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group l…
In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\infty$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\infty$-algebra. Using the controlling algebra ap…
We describe $L_\infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_\infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronov's higher derived brackets construction which is of independent interest.
Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order $n$ deformations to order $n+1$ deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorp…
