hal-03286237
https://universite-paris-saclay.hal.science/hal-03286237
https://universite-paris-saclay.hal.science/hal-03286237/document
https://universite-paris-saclay.hal.science/hal-03286237/file/cohomology-algorithm.pdf
doi:10.1007/s00006-011-0319-z
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-PSUD] Université Paris Sud - Paris XI
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[LM-ORSAY] Laboratoire de Mathématiques d'Orsay
[UNIV-PARIS-SACLAY] Université Paris-Saclay
[UNIV-PSUD-SACLAY] Université Paris Sud pour Paris Saclay
[GS-MATHEMATIQUES] Graduate School Mathématiques
A Gröbner-Bases Algorithm for the Computation of the Cohomology of Lie (Super) Algebras
Aghasi, Mansour
M.-Alizadeh, Benyamin
Merker, Joël
Sabzevari, Masoud
[MATH] Mathematics [math]
ART
Lie (super) algebras
cohomology
reduced Gröbner bases
quotient vector spaces
homogeneity splitting
Lie (super) algebras
We 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 Z^k / B^k of cocycles Z^k modulo coboundaries B^k , 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 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.
2011-12-01
2021-07-17
en
Advances in Applied Clifford Algebras
Springer Verlag