计算李（超）代数上同调的Groebner基算法
A Groebner-bases algorithm for the computation of the cohomology of Lie (super) algebras
摘要：
给出了特征为零的可换域K上生成李（超）代数的标准上同调空间的一个有效算法。为了得到上圈Z^k模上边界B^k的商空间Z^k/B^k的某些生成元的显式表示，我们应用Groebner基技术（在适当的线性设置中）并利用它们的强度。此外，当所考虑的李（超）代数具有分次时--这种情况在表示论和微分几何中都经常发生--所有的上同调空间Z^k/B^k自然地分裂为更小的子空间的直和，这使我们能够在高维李（超）代数中提高计算机的计算速度。最后，我们在Maple软件中实现了我们的算法，并通过一些例子来评估它的性能，其中大部分在Cartan-Tanaka连接理论中有一些应用。
英文摘要：
We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a commutative field K of characteristic zero. In order to reach explicit representatives of some generators of the quotient space Z^k/B^k of cocycles Z^k modulo coboundaries B^k, we apply Groebner bases techniques (in the appropriate linear setting) and take advantage of their strength. Moreover, when the considered Lie (super) algebras enjoy a grading -- a case which often happens both in representation theory and in differential geometry --, all cohomology spaces Z^k/B^k naturally split up as direct sums of smaller subspaces, and this enables us, for higher dimensional Lie (super) algebras, to improve the computer speed of calculations. Lastly, we implement our algorithm in the Maple software and evaluate its performances via some examples, most of which have several applications in the theory of Cartan-Tanaka connections.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
