Deformation maps of Quasi-twilled associative algebras

Shanshan Liu, Abdenacer Makhlouf, Lina Song
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Abstract

In this paper, we introduce two types of deformation maps of quasi-twilled associative algebras. Each type of deformation maps unify various operators on associative algebras. Right deformation maps unify modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms and crossed homomorphisms. Left deformation maps unify relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of associative algebras. Furthermore, we give the controlling algebra and the cohomology of these two types of deformation maps. On the one hand, we obtain some existing results for modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms, crossed homomorphisms, relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators and Reynolds operators. On the other hand, we also obtain some new results, such as the controlling algebra of a modified Rota-Baxter operator of weight $\lambda$ on an associative algebra, the controlling algebra and the cohomology of a deformation map of a matched pair of associative algebras.
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准扭曲关联代数的变形映射
在本文中,我们介绍了两类准踌躇关联代数的变形映射。每种类型的变形映射都统一了共轭代数上的各种算子。右变形映射统一了权重为 $\lambda$ 的修正罗塔-巴克斯特算子、派生、同态和交叉同态。左变形映射统一了权重为 0 的相对罗塔-巴克斯特算子、扭曲罗塔-巴克斯特算子、雷诺算子和关联代数的配对变形映射。此外,我们还给出了这两类变形映射的控制代数和同调。一方面,我们得到了关于权重为 $\lambda$ 的修正罗塔-巴克斯特算子、衍生、同态、交叉同态、权重为 0 的相对罗塔-巴克斯特算子、扭曲罗塔-巴克斯特算子和雷诺算子的一些已有结果。另一方面,我们还得到了一些新的结果,如关联代数上权重为 $\lambda$ 的修正 Rota-Baxter 算子的控制代数、一对匹配关联代数的变形映射的控制代数和同调。
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