论文标题
弱霍普夫代数捕获的代数的对称性
Symmetries of algebras captured by actions of weak Hopf algebras
论文作者
论文摘要
在本文中，我们介绍了关于$ \ bbbk $ -Algebras对称的良好结果的概括，其中$ \ bbbk $是一个字段。传统上，对于$ \ bbbk $ -Algebra $ a $，$ \ bbbk $ -Algebra $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $。同样，$ a $的派生的谎言代数捕获了$ a $ a $ a $ a $ a $ a $的代数。在本文中，给定一个类别$ \ MATHCAL {C} $，其对象具有$ \ bbbk $ -linear单型模块类别类别，我们介绍了一个对象$ \ operatorName {sym} _ {\ Mathcal {c}}}（c}}（c}}}（a）$ cape $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a的symmetries。我们的研究涵盖了各种类别，其物体包括群体素，谎言代数和更普遍的同时性弱霍普夫代数。值得注意的是，我们证明，对于一个积极分级的非连接$ \ bbbk $ -Algebra $ a $，其某些对称性自然会在弱的Hopf框架内捕获。
In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework.
