准扭曲关联代数、变形映射及其支配代数

Apurba Das, Ramkrishna Mandal
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摘要

准凋摆关联代数是一个关联代数 $/mathbb{A}$,它的底层向量空间有一个分解 $/mathbb{A} = A \oplus B$,使得 $B \subset \mathbb{A}$ 是一个子代数。在本文的第一部分,我们给出了毛勒-卡尔坦特征,并介绍了准凋零关联代数的同调。在一个准凋零关联代数 $\mathbb{A}$ 中,如果 $\mathrm{Gr}(D)\subset\mathbb{A}$ 是一个子代数,那么线性映射 $D: A\rightarrow B$ 就被称为强变形映射。这样的映射概括了关联代数同态、派生、交叉同态以及修正{\sf r}-矩阵的关联类似。我们引入了强变形映射 $D$ 的同调,它统一了上述所有算子的同调。我们还定义了一对$(\mathbb{A}, D)$的支配代数,以研究$\mathbb{A}$和$D$的同时变形。另一方面,线性映射 $r:如果 $\mathrm{Gr} (r) \subset \mathbb{A}$ 是一个子代数,那么 B \rightarrow A$ 就叫做弱变形映射。这样的映射泛化了任意权重的相对罗塔-巴克斯特算子、扭曲罗塔-巴克斯特算子、雷诺算子、左平均算子和右平均算子。在这里,我们定义了弱变形映射 $r$ 的同调与支配代数(统一了上述所有算子的同调),以及支配同时变形的一对 $(\mathbb{A}, r)$ 的同调与支配代数。
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Quasi-twilled associative algebras, deformation maps and their governing algebras
A quasi-twilled associative algebra is an associative algebra $\mathbb{A}$ whose underlying vector space has a decomposition $\mathbb{A} = A \oplus B$ such that $B \subset \mathbb{A}$ is a subalgebra. In the first part of this paper, we give the Maurer-Cartan characterization and introduce the cohomology of a quasi-twilled associative algebra. In a quasi-twilled associative algebra $\mathbb{A}$, a linear map $D: A \rightarrow B$ is called a strong deformation map if $\mathrm{Gr}(D) \subset \mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra homomorphisms, derivations, crossed homomorphisms and the associative analogue of modified {\sf r}-matrices. We introduce the cohomology of a strong deformation map $D$ unifying the cohomologies of all the operators mentioned above. We also define the governing algebra for the pair $(\mathbb{A}, D)$ to study simultaneous deformations of both $\mathbb{A}$ and $D$. On the other hand, a linear map $r: B \rightarrow A$ is called a weak deformation map if $\mathrm{Gr} (r) \subset \mathbb{A}$ is a subalgebra. Such a map generalizes relative Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators, left-averaging operators and right-averaging operators. Here we define the cohomology and governing algebra of a weak deformation map $r$ (that unify the cohomologies of all the operators mentioned above) and also for the pair $(\mathbb{A}, r)$ that govern simultaneous deformations.
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