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تسجيل مستخدم جديدDeformations and homotopy theory of relative Rota-Baxter Lie algebras
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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. Next, the notion of a emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to emph{pre-Lie$_infty$-algebras}.
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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 Voronovs higher derived brackets construction which is of independent interest.
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 level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.
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-L
ieRep 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 cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.
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 relat
ive 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 Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.
In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight $1$. First we recall the category of relative Rota-Baxter operators of weight $1$ on Lie algebras and construct a cohomology theory for them. We use the secon
d cohomology group to study infinitesimal deformations of relative Rota-Baxter operators and modified $r$-matrices. Then we introduce a cohomology theory of relative Rota-Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota-Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving Van Est theorems between the two cohomology theories. Finally, we integrate a relative Rota-Baxter operator of weight 1 on a Lie algebra to a local relative Rota-Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other.
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