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TitleA Gr\"obner-bases algorithm for the computation of the cohomology of Lie (super) algebras.
Author(s) Mansour Aghasi, Benyamin M.-Alizadeh, Joël Merker, Masoud Sabzevari
TypeArticle in Journal
AbstractWe present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a field 𝕂 of characteristic zero. In order to reach explicit representatives of some generators of the quotient space 𝒵k/ℬk of cocycles 𝒵k modulo coboundaries ℬk , we apply Gröbner 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 𝒵k/ℬk 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.
KeywordsLie (super) algebras, cohomology, reduced Gröbner bases, quotient vector spaces, homogeneity splitting
ISSN0188-7009; 1661-4909/e
URL http://link.springer.com/article/10.1007%2Fs00006-011-0319-z
JournalAdv. Appl. Clifford Algebr.
PublisherSpringer (Birkh\"auser), Basel
Translation No
Refereed No