Abstract
A hom-Lie algebroid is a vector bundle together with a Lie algebroid-like structure which is twisted by a homomorphism and a VB-hom algebroid is essentially defined as a hom-Lie algebroid object in the category of vector bundles. In this paper, we show that, there exists a correspondence between the VB-hom algebroids and two term representations up to homotopy of hom-Lie algebroid.
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References
Abad CA, Crainic M (2012) Representations up to homotopy of Lie algebroids. J Reine Angew Math 663:91–126
Cai L, Liu J, Sheng Y (2017) Hom–Lie algebroids, Hom–Lie bialgebroids and Hom–Courant algebroids. J Geom Phys 121:15–32
Chen Z, Liu Z, Sheng Y (2014) On double vector bundles. Acta Math Sinica 30(10):1655–1673
Crainic M, Fernandes RL (2011) Lectures on integrability of Lie brackets. Geom Topol Monogr 17:1–107
Gracia-Saz A, Mehta RA (2010) Lie algebroid structures on double vector bundles and representation theory of Lie algebroids. Adv Math 223(4):1236–1275
Hartwig J, Larsson D, Silvestrov S (2006) Deformations of Lie algebras using \(\sigma \)-derivations. J Algebra 295:314–361
Hassanzadeh M (2019) Lagrange’s theorem for Hom-groups. Rocky Mountain J. Math. 49(3):773–787
Jiang J, Mishra SK, Sheng Y (2019) Hom-Lie algebras and Hom-Lie groups, integration and differentiation. arXiv:1904.06515v1 [math.GR]
Laurent-Gengoux C, Teles J (2013) Hom-Lie algebroids. J Geom Phys 68:69–75
Mackenzie KCH (1992) Double Lie algebroids and second-order geometry I. Adv Math 94(2):180–239
Mackenzie KCH (2000) Double Lie algebroids and second-order geometry II. Adv Math 154:46–75
Mackenzie KCH (2005) General theory of Lie groupoids and Lie algebroids. London mathematical society lecture note series. Cambridge University Press, Cambridge, p 213
Mackenzie KCH (2011) Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids. J Reine Angew Math 658:193–245
Makhlouf A, Silvestrov S (2008) Hom-algebra structures. J Gen Lie Theory Appl 2(2):51–64
Mehta RA (2014) Lie algebroid modules and representations up to homotopy. Indag Math 25:1122–1134
Merati S, Farhangdoost MR (2018a) Representation and central extension of Hom–Lie algebroids. J Algebra Appl 17(11):185219
Merati S, Farhangdoost MR (2018b) Representation up to homotopy of Hom–Lie algebroids. Int J Geom Methods Mod Phys 15(5):1850074
Pradines J (1968) Géométrie differentielle au-dessus d’un grupoïde. C R Acad Sci Paris Série A 266:1194–1196
Sheng Y (2012a) Representations of Hom–Lie algebras. Algebr Represent Theor 15:1081–1098
Sheng Y (2012b) On deformations of Lie algebroids. Results Math 62:103–120
Sheng Y, Zhu C (2011) Semidirect products of representations up to homotopy. Pac J Math 249(1):211–236
Vaǐntrob AY (1997) Lie algebroids and homological vector fields. Uspekhi Mat Nauk 52(2(314)):161–162
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This work was funded by Shiraz University.
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Merati, S., Farhangdoost, M.R. VB-Hom Algebroid Morphisms and 2-Term Representation Up to Homotopy of Hom-Lie Algebroids. Iran J Sci Technol Trans Sci 45, 937–944 (2021). https://doi.org/10.1007/s40995-020-01049-1
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DOI: https://doi.org/10.1007/s40995-020-01049-1