Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T00:35:16.726Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Lie Algebraic Approach to Abelian Extensions of Associative Algebras

Part of: Rings and algebras with additional structure Modules, bimodules and ideals Homological methods

Published online by Cambridge University Press:  26 March 2020

Youjun Tan  and
Senrong Xu
Youjun Tan
Affiliation:
College of Mathematics, Sichuan University, Chengdu610064, China e-mail: [email protected]
Senrong Xu*
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang 212013, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

Keywords

Bimodulescohomologiesderivationsabelian extensions

MSC classification

Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 16W25: Derivations, actions of Lie algebras 16D20: Bimodules
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 25 - 38
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Senrong Xu is the corresponding author.

References

Baer, R., Erweiterung von Gruppen und ihren Isomorphismen. Math. Z. 38(1934), 375–416. https://doi.org/10.1007/BF01170643Google Scholar
Bardakov, V. G. and Singh, M., Extensions and automorphisms of Lie algebras. J. Algebra Appl. 16(2017), 1750162. https://doi.org/10.1142/S0219498817501626CrossRefGoogle Scholar
Hilton, P. J. and Stammbach, U., A course in homological algebra. 2nd ed. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4419-8566-8Google Scholar
Hochschild, G. P., On the cohomology groups of an associative algebra. Ann. of Math. 46(1945), 58–67. https://doi.org/10.2307/1969145CrossRefGoogle Scholar
Jacobson, N., The theory of rings. Math. Surv. No. II, Amer. Math. Soc., 1943.Google Scholar
Jin, P. and Liu, H., The Wells exact sequence for the automorphism group of a group extension. J. Algebra 324(2010), 1219–1228. https://doi.org/10.1016/j.jalgebra.2010.04.034CrossRefGoogle Scholar
Loday, J.-L., Cyclic homology. Grundlehren der mathematischen Wissenschaften 301, 2nd ed., Springer, 1998.Google Scholar
Passi, I. B. S., Singh, M., and Yadav, M. K., Automorphisms of abelian group extensions. J. Algebra 234(2010), 820–830. https://doi.org/10.1016/j.jalgebra.2010.03.029Google Scholar
Pierce, R. S., Associative algebras. Springer-Verlag, New York Inc., 1982.CrossRefGoogle Scholar
Robinson, D. J. S., Automorphisms of group extensions. Note Mat. 33(2013), 121–129.Google Scholar
Tan, Y. and Xu, S., The Wells map for abelian extensions of 3-Lie algebras. Czech. Math J. 69(2019), 1133–1164. https://doi.org/10.21136/CMJ.2019.0098-18CrossRefGoogle Scholar
Weibel, C. A., An introduction to homological algebra. Cambridge University Press, 1994.CrossRefGoogle Scholar
Wells, C., Automorphisms of group extensions. Trans. Amer. Math. Soc. 155(1971), 189–194. https://doi.org/10.2307/1995472CrossRefGoogle Scholar
Xu, S., Cohomology, derivations and abelian extensions of 3-Lie algebras. J. Algebra Appl. 18(2019), 1950130, 26 pp. https://doi.org/10.1142/S0219498819501305Google Scholar