标题： 李超代数的新超级双导子定义 显示英文标题

标题： A New Definition of Superbiderivations for Lie Superalgebras

摘要： 在本文中，我们从结构和几何的角度研究李超代数上的优双导子。 受经典事实的启发，即李代数的括号本身是一个双导子，我们为李超代数提出了一个优双导子的新定义，该定义要求括号是一个优双导子，而现有的文献中的定义并不满足这一条件。 我们的重点是完备李超代数，这是半单李代数的一个自然推广，在寻找替代结构概念的过程中已成为一个有前景的框架。 在此背景下，我们引入并研究了线性超交换映射，并将我们的定义与之前的提议进行了比较。 最后，我们给出了两个应用：一个是涉及海森堡李超代数的超导子超代数，另一个是通过优双导子提供变形理论的初步几何见解。 显示英文摘要

摘要： In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations.