论文标题
Duflo-serganova同源性，用于与cartan矩阵的特殊模块化超级甲壳虫
Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix
论文作者
论文摘要
对于特殊的有限维模块化谎言superalgebras $ \ mathfrak {g}（a）$，带有不可分解的cartan矩阵$ a $及其简单的分子质量，我们计算了构成具有零广场奇数元素的同源物的非同构lie superalgebras。这些同源性是代表理论的Duflo-Serganova方法中的关键要素。 文献中具有不同应用范围的文献中，有两个定义的谎言超级甲虫的缺陷定义。我们建议第三个定义和易于使用的方法来找到其价值。 在积极的特征中，我们发现了一个又一个理由考虑与重量空间不同的重量空间，应在地面上考虑。 我们证明，应在伴随模块中考虑同源元素的等级（在计算给定谎言超级骨的缺陷方面具有决定性的等级），而不是最小尺寸的不可约的模块（尽管有时可以考虑后者，例如，$ p = 0 $）。 我们还计算了上述同源性的唯一一个简单的谎言超级甲壳虫的情况，其对称根系的复数及其模块化版本及其模块化版本及其模块化。
For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more reason to consider the space of roots over reals, unlike the space of weights, which should be considered over the ground field. We proved that the rank of the homological element (decisive in calculating the defect of a given Lie superalgebra) should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for $p=0$). We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its modular versions: $\mathfrak{psl}(a|a+pk)$ for $a$ and $k$ small, and $p=2, 3, 5$.
