A Review on Hom-Gerstenhaber Algebras and Hom-Lie Algebroids

  • Satyendra Kumar MishraEmail author
  • Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)


The aim of the present article is to review the current progress on Hom-Gerstenhaber algebras and Hom-Lie algebroids. There are two different definitions of Hom-Lie algebroids. The modification was made in the original definition to consider some essential results on Hom-Lie algebroids and discuss some new examples. However, it turns out that for such results such a modification is not required. There are several attempts on defining representations and cohomology of Hom-Lie algebroids. As expected from the case of Hom-Lie algebras, there is no unique cohomology for Hom-Lie algebroids. We discuss about representations and cohomology of Hom-Lie algebroids that yields a differential calculus and dual description for Hom-Lie algebroids. Later on, we summarise the relationship between Hom-Lie algebroids and Hom-Gerstenhaber algebras.


Hom-Gerstenhaber algebras Hom-Lie algebroids Hom-Lie algebras 

MSC 2010 Classification

17B75 17B99 17B35 17B37 17D99 16W10 16W55 



Satyendra Kumar Mishra is grateful to the research environment in Mathematics and Applied Mathematics (MAM), Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University, Västerås, Sweden for providing support and excellent research environment during his visit to Mälardalen University when part of the work on this paper has been performed.


  1. 1.
    Abad, C.A., Crainic, M.: Representations up to homotopy of Lie algebroids. J. Reine Angew. Math. 663, 91–126 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aizawa, N., Sato, H.-T.: \(q\)-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256(2), 185–190 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras. J. Lie Theory 21(4), 813–836 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76, 38–60 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Caenepeel, S., Goyvaerts, I.: Monoidal Hom-Hopf Algebras. Commun. Algebra 39(6), 2216–2240 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cai, L., Sheng, Y.: Purely Hom-Lie bialgebra. Sci. China Math. (2018)Google Scholar
  7. 7.
    Cai, L., Liu, J., Sheng, Y.: Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids. J. Geom. Phys. 121, 15–32 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chaichian, M., Kulish, P., Lukierski, J.: \(q\)-deformed Jacobi identity, \(q\)-oscillators and \(q\)-deformed infinite-dimensional algebras. Phys. Lett. B 237, 401–406 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chaichian, M., Isaev, A.P., Lukierski, J., Popowicz, Z., Prešnajder, P.: \(q\)-deformations of Virasoro algebra and conformal dimensions. Phys. Lett. B 262(1), 32–38 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chakrabarti, R., Jagannathan, R.: A \((p, q)\)-deformed Virasoro algebra. J. Phys. A 25(9), 2607–2614 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Crainic, M., Fernandes, R.L.: Lectures on integrability of Lie brackets. Geom. Topol. Monogr. 17, 1–107 (2011)Google Scholar
  12. 12.
    Elchinger, O.: Formalité liée aux algébres enveloppantes et étude des algébres Hom-(co)Poisson. Thése de Doctorat, Université de Haute Alsace, November (2012)Google Scholar
  13. 13.
    Gerstenhaber, M., Schack, S.D.: Algebras, bialgebras, quantum groups and algebraic deformations. In: Gerstenhaber M., Stasheff J. (eds.), Deformations Theory and Quantum Groups with Applications to Mathematical Physics. Contemporary Mathematics, vol. 134, 51–92. AMS, Providence (1992)Google Scholar
  14. 14.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79(2), 59–103 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gerstenhaber, M.: On the deformation of rings and algebras II. Ann. Math. 84(2), 1–19 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gerstenhaber, M.: On the deformation of rings and algebras III. Ann. Math. 88(2), 1–34 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gerstenhaber, M.: On the deformation of rings and algebras IV. Ann. Math. 99(2), 257–276 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hartwig, J.T., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using \(\sigma -\)derivations. J. Algebra 295, 314–361 (2006) (Preprint in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, 52 pp. (2003))Google Scholar
  20. 20.
    Hellström, L., Makhlouf, A., Silvestrov, S.D.: Universal algebra applied to hom-associative algebras, and more. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds.), Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, vol. 85, 157–199. Springer, Berlin, Heidelberg (2014)Google Scholar
  21. 21.
    Huebschmann, J.: Duality for Lie-Rinehart algebras and the modular class. J. Reine Angew. Math. 510, 103–159 (1999)Google Scholar
  22. 22.
    Huebschmann, J.: Lie-Rinehart Algebras, Descent, and Quantization. In: Janelidze, G., Pareigis, B. Tholen, W. (eds.), Galois theory, Hopf algebras, and semiabelian categories. Fields Institute Communications, vol. 43, 295–316. American Mathematical Society, Providence, RI (2004)Google Scholar
  23. 23.
    Huebschmann, J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57–113 (1990)Google Scholar
  24. 24.
    Huebschmann, J.: Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble) 48, 425–440 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Huebschmann, J.: Extensions of Lie-Rinehart algebras and cotangent bundle reduction. Proc. Lond. Math. Soc. 107(5), 1135–1172 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Huebschmann, J.: Multi derivation Maurer-Cartan algebras and sh Lie-Rinehart algebras. J. Algebra 472, 437–479 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Larsson, D., Silvestrov, S.D.: Quasi-Hom-Lie algebras, central extensions and \(2\)-cocycle-like identities. J. Algebra 288, 321–344 (2005) (Preprints in Mathematical Sciences 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2004))Google Scholar
  28. 28.
    Larsson, D., Silvestrov, S.D.: Quasi-Lie algebras. In: Noncommutative Geometry and Representation Theory in Mathematical Physics. Contemporary Mathematics, vol. 391, 241–248. American Mathematical Society, Providence, RI (2005) (Preprints in Mathematical Sciences 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2004))Google Scholar
  29. 29.
    Larsson, D., Silvestrov, S.D.: Graded quasi-Lie agebras. Czechoslov. J. Phys. 55, 1473–1478 (2005)CrossRefGoogle Scholar
  30. 30.
    Larsson, D., Silvestrov, S.D.: Quasi-deformations of \(sl_2(\mathbb{F})\) using twisted derivations. Commun. Algebra 35, 4303–4318 (2007)zbMATHCrossRefGoogle Scholar
  31. 31.
    Larsson, D., Sigurdsson, G., Silvestrov, S.D.: Quasi-Lie deformations on the algebra \(\mathbb{F}[t]/(t^N)\). J. Gen. Lie Theory Appl. 2, 201–205 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Laurent-Gengoux, C., Teles, J.: Hom-Lie algebroids. J. Geom. Phys. 68, 69–75 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Laurent-Gengoux, C., Makhlouf, A., Teles, J.: Universal algebra of a Hom-Lie algebra and group-like elements. J. Pure Appl. Algebra 222(5), 1139–1163 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids. Lecture Note Series, vol. 213. London Mathematical Society (2005)Google Scholar
  35. 35.
    Mackenzie, K.: Double Lie algebroids and second-order geometry I. Adv. Math. 94(2), 180–239 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008) (Preprints in Mathematical Sciences 2006:10, LUTFMA-5074-2006, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2006))Google Scholar
  37. 37.
    Makhlouf, A., Silvestrov, S.: Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 189–206 (2009)Google Scholar
  38. 38.
    Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras. Int. Electron. J. Algebra 8, 177–190 (2010)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Makhlouf, A., Silvestrov, S.: Hom-algebras and Hom-coalgebras. J. Algebra Its Appl. 9, 553–589 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Makhlouf, A., Silvestrov, S.: Notes on \(1\)-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–759 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Mandal, A., Mishra, S.K.: Hom-Lie-Rinehart algebras. Commun. Algebra 46(9), 3722–3744 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Mandal, A., Mishra, S.K.: On Hom-Gerstenhaber algebras, and Hom-Lie algebroids. J. Geom. Phys. 133, 287–302 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Mehta, R.A.: Lie algebroid modules and representations up to homotopy. Indag. Math. 25, 1122–1134 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Merati, S., Farhangdoost, M.R.: Representation up to homotopy of Hom-Lie algebroids. Int. J. Geom. Methods Mod. Phys. 15(5), 1850074 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Merati, S., Farhangdoost, M.R.: Representation and central extension of Hom-Lie algebroids. J. Algebra Its Appl. 17(11), 1850219 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Nakamura, T.: Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids. arXiv:1907.05004v1 [math.SG]
  47. 47.
    Peyghan, E., Baghban, A., Sharahi, E.: Linear Poisson structures and Hom-Lie algebroids. Revista De La Unión Math. Argent. 60(2), 299–313 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Polychronakos, A.P.: Consistency conditions and representations of a \(q\)-deformed Virasoro algebra. Phys. Lett. B 256(1), 35–40 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Richard, L., Silvestrov, S.D.: Quasi-Lie structure of \(\sigma \)-derivations of \(\mathbb{C}[t^{\pm 1}]\). J. Algebra 319(3), 1285–1304 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Sato, H.T.: Realizations of \(q\)-deformed Virasoro algebra. Prog. Theor. Phys. 89(2), 531–544 (1993)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Sato, H.T.: \(q\)-Virasoro operators from an analogue of the Noether currents. Z. Phys. C 70(2), 349–355 (1996)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Sato, H.T.: OPE formulae for deformed super-Virasoro algebras. Nucl. Phys. B 471, 553–569 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Schwarzbach, Y.K.: Exact Gerstenhaber algebras and Lie bialgebroid. Acta Appl. Math. 41, 153–165 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Sheng, Y.: Representation of Hom-Lie algebras. Algebr. Reprensent. Theory 15(6), 1081–1098 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Sheng, Y., Xiong, Z.: On Hom-Lie algebras. Linear Multilinear Algebra 63(6), 2379–2395 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type. Czechoslov. J. Phys. 56, 1287–1291 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Sigurdsson, G., Silvestrov, S.: Lie color and Hom-Lie algebras of Witt type and their central extensions. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 247–255. Springer, Berlin (2009)Google Scholar
  58. 58.
    Vaintrob, A.Y.: Lie algebroids and homological vector fields. Uspekhi Mat. Nauk 52(2(314)), 161–162 (1997)Google Scholar
  59. 59.
    Xiong, Z.: Equivalent description of Hom-Lie algebroids. Adv. Math. Phys. Article ID 8417516, 8 pp. (2018)Google Scholar
  60. 60.
    Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200, 545–560 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Yau, D.: Hom-algebras and homology. J. Lie Theory 19(2), 409–421 (2009)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Yau, D.: Hom-bialgebras and comodule Hom-algebras. Int. Electron. J. Algebra 8, 45–64 (2010)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Zhang, Q., Yu, H., Wang, C.: Hom-Lie algebroids and hom-left-symmetric algebroids. J. Geom. Phys. 116, 187–203 (2017)MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics UnitIndian Statistical Institute, Bangalore CentreBangaloreIndia
  2. 2.Division of Applied MathematicsSchool of Education, Culture and Communication, Mälardalen UniversityVästeråsSweden

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