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Register a new userRota-Baxter operators of nonzero weight on the matrix algebra of order three
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We classify all Rota-Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.
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We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group algebra of a group $G$ or $H=U(mathfrak{g})$ the universal enveloping algebra of a Lie algebra $mathfrak{g}$, then we prove that Rota-Baxter operators on $H$ are in one to one correspondence with corresponding Rota-Baxter operators on groups or Lie algebras.
An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota-Baxter operator and how this is related to the ordered monoid.
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 second 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.
In this paper, we first construct a graded Lie algebra which characterizes Rota-Baxter operators on an anti-flexible algebra as Maurer-Cartan elements. Next, we study infinitesimal deformations of bimodules over anti-flexible algebras. We also consider compatible Rota-Baxter operators on bimodules over anti-flexible algebras. Finally, We define $mathcal{ON}$-structures which give rise to compatible Rota-Baxter operators and vice-versa.
In the paper we study homogeneous Rota-Baxter operators with weight zero on the infinite dimensional simple $3$-Lie algebra $A_{omega}$ over a field $F$ ( $ch F=0$ ) which is realized by an associative commutative algebra $A$ and a derivation $Delta$ and an involution $omega$ ( Lemma mref{lem:rbd3} ). A homogeneous Rota-Baxter operator on $A_{omega}$ is a linear map $R$ of $A_{omega}$ satisfying $R(L_m)=f(m)L_m$ for all generators of $A_{omega}$, where $f : A_{omega} rightarrow F$. We proved that $R$ is a homogeneous Rota-Baxter operator on $A_{omega}$ if and only if $R$ is the one of the five possibilities $R_{0_1}$, $R_{0_2}$,$R_{0_3}$,$R_{0_4}$ and $R_{0_5}$, which are described in Theorem mref{thm:thm1}, mref{thm:thm4}, mref{thm:thm01}, mref{thm:thm03} and mref{thm:thm04}. By the five homogeneous Rota-Baxter operators $R_{0_i}$, we construct new $3$-Lie algebras $(A, [ , , ]_i)$ for $1leq ileq 5$, such that $R_{0_i}$ is the homogeneous Rota-Baxter operator on $3$-Lie algebra $(A, [ , , ]_i)$, respectively.
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