Extensions of hom-Lie algebras in terms of cohomology

Abdoreza R. Armakan; Mohammed Reza Farhangdoost

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 317-328
  • ISSN: 0011-4642

Abstract

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We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra 𝔤 by another hom-Lie algebra 𝔥 and discuss the case where 𝔥 has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology.

How to cite

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Armakan, Abdoreza R., and Farhangdoost, Mohammed Reza. "Extensions of hom-Lie algebras in terms of cohomology." Czechoslovak Mathematical Journal 67.2 (2017): 317-328. <http://eudml.org/doc/288215>.

@article{Armakan2017,
abstract = {We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $\mathfrak \{g\}$ by another hom-Lie algebra $\mathfrak \{h\}$ and discuss the case where $\mathfrak \{h\}$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology.},
author = {Armakan, Abdoreza R., Farhangdoost, Mohammed Reza},
journal = {Czechoslovak Mathematical Journal},
keywords = {hom-Lie algebras; cohomology of hom-Lie algebras; extensions of hom-Lie algebras},
language = {eng},
number = {2},
pages = {317-328},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extensions of hom-Lie algebras in terms of cohomology},
url = {http://eudml.org/doc/288215},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Armakan, Abdoreza R.
AU - Farhangdoost, Mohammed Reza
TI - Extensions of hom-Lie algebras in terms of cohomology
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 317
EP - 328
AB - We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $\mathfrak {g}$ by another hom-Lie algebra $\mathfrak {h}$ and discuss the case where $\mathfrak {h}$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology.
LA - eng
KW - hom-Lie algebras; cohomology of hom-Lie algebras; extensions of hom-Lie algebras
UR - http://eudml.org/doc/288215
ER -

References

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