对称单模李（超）代数的形变
Deformations of Symmetric Simple Modular Lie (Super)Algebras
AI总结：
本研究在正特征代数闭域上计算了秩<9的有限维对称单李（超）代数无穷小形变的上同调，发现奇上循环对应的形变均可积且为全新结果，给出了特定特征下若干代数的形变上循环分类，并提出相关开放问题。
摘要：
我们称一个李（超）代数是“对称的”，若相对于极大环面的每个根，都存在重数相同的相反根。在特征为正的代数闭域上（特征最高至7或11，足以提出一般性猜想），我们计算了所有秩小于9的已知有限维对称单李（超）代数的无穷小形变对应的上同调，仅部分考虑了带ADE根系的李代数的超化以及 queer 化李代数。任何李超代数的形变模构成一个超簇。由任意奇上循环给出的无穷小形变都是可积的，所有对应奇上循环的形变都是全新的。新成果包括：特征3下29维Brown代数的形变（形变结果）的上循环分类，以及特征2下Weisfeiler-Kac代数和无Cartan矩阵的正交李代数的上循环分类。待解决问题：刻画非同构形变以及上同调理论的等价类。附录：针对复单李代数的若干模类比，以及特征3和2下特有的单李代数，我们刻画了带平凡系数的上同调空间，发现该空间中的自然乘法结构极为复杂。
英文摘要：
We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank $<9$, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.
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