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Register a new userHomogeneous Rota-Baxter operators on $A_{omega}$ (II)
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In this paper we study $k$-order homogeneous Rota-Baxter operators with weight $1$ on the simple $3$-Lie algebra $A_{omega}$ (over a field of characteristic zero), which is realized by an associative commutative algebra $A$ and a derivation $Delta$ and an involution $omega$ (Lemma mref{lem:rbd3}). A $k$-order homogeneous Rota-Baxter operator on $A_{omega}$ is a linear map $R$ satisfying $R(L_m)=f(m+k)L_{m+k}$ for all generators ${ L_m~ |~ min mathbb Z }$ of $A_{omega}$ and a map $f : mathbb Z rightarrowmathbb F$, where $kin mathbb Z$. We prove that $R$ is a $k$-order homogeneous Rota-Baxter operator on $A_{omega}$ of weight $1$ with $k eq 0$ if and only if $R=0$ (see Theorems 3.2, and $R$ is a $0$-order homogeneous Rota-Baxter operator on $A_{omega}$ of weight $1$ if and only if $R$ is one of the forty possibilities which are described in Theorems3.5, 3.7, 3.9, 3.10, 3.18, 3.21 and 3.22.
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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.
Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension $leq 3$ and their corresponding pre-Lie algebras.
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 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.
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