Abstract
We investigate the conditions under which the smash product of an (ordinary or restricted) enveloping algebra and a group algebra is Lie solvable or Lie nilpotent.
Similar content being viewed by others
References
Yu. Bahturin and V. Petrogradsky, Polynomial identities in smash products, Journal of Lie Theory 12 (2002), 369–395.
J. Bergen, D. M. Riley and H. Usefi, Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity, Israel Journal of Mathematics 196 (2013), 161–173.
A. K. Bhandari and I. B. S. Passi, Lie nilpotency indices of group algebras, Bulletin of the London Mathematical Society 24 (1992), 68–70.
A. Bovdi and A. Grishkov, Lie properties of crossed products, Journal of Algebra 320 (2008), 3447–3460.
A. A. Bovdi and I. I. Khripta, Generalized Lie nilpotent group rings, Mathematics of the U.S.S.R. Sbornik 57 (1987), 165–169.
A. A. Bovdi and J. Kurdics, Lie properties of the group algebra and the nilpotency class of the group of units, Journal of Algebra 212 (1999), 28–64.
S. Dăscălescu, C. Năstăsescu and S. Raianu, Hopf Algebras. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 235, Marcel Dekker, New York, 2001.
N. D. Gupta and F. Levin, On the Lie ideals of a ring, Journal of Algebra 81 (1983), 225–231.
N. Jacobson, Lie Algebras, Dover, New York, 1979.
S. A. Jennings, Central chains of ideals in an associative ring, Duke Mathematical Journal 9 (1942), 341–355.
M. Kochetov, PI Hopf algebras of prime characteristic, Journal of Algebra 262 (2003), 77–98.
S. Montgomery, Hopf Algebras and their Actions on Rings, CMBS Regional Conference Series in Mathematics, Vol. 82, American Mathematical Society, Providence, RI, 1993.
I. B. S. Passi, D. S. Passman and S. K. Sehgal, Lie solvable group rings, Canadian Journal of Mathematics 25 (1973), 748–757.
D. M. Riley, PI-algebras generated by nilpotent elements of bounded index, Journal of Algebra 192 (1997), 1–13.
D. M. Riley and A. Shalev, The Lie structure of enveloping algebras, Journal of Algebra 162 (1993), 46–61.
D. M. Riley and A. Shalev, Restricted Lie algebras and their envelopes, Canadian Journal of Mathematics 47 (1995), 146–164.
D. M. Riley and V. Tasić, Lie identities for Hopf algebras, Journal of Pure and Applied Algebra 122 (1997), 127–134.
S. K. Sehgal, Topics in Group Rings, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 50, Marcel Dekker, New York, 1978.
A. Shalev, The derived length of Lie soluble group rings. I, Journal of Pure and Applied Algebra 78 (1992), 291–300.
A. Shalev, The derived length of Lie soluble group rings. II, Journal of the London Mathematical Society 49 (1994), 93–99.
R. K. Sharma and J. B. Srivastava, Lie solvable rings, Proceedings of the American Mathematical Society 94 (1985), 1–8.
S. Siciliano, Lie derived lengths of restricted universal enveloping algebras, Publicationes Mathematicae Debrecen 68 (2006), 503–513.
S. Siciliano, On Lie solvable restricted enveloping algebras, Journal of Algebra 314 (2007), 226–234.
S. Siciliano and H. Usefi, Lie solvable enveloping algebras of characteristic two, Journal of Algebra 382 (2013), 314–331.
H. Strade, Simple Lie Algebras over Fields of Positive Characteristic I. Structure Theory, de Gruyter Expositions in Mathematics, Vol. 38, Walter de Gruyter & Co., Berlin, 2004.
H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 116, Marcel Dekker, New York, 1988.
H. Usefi, Lie identities on enveloping algebras of restricted Lie superalgebras, Journal of Algebra 393 (2013), 120–131.
E. Zalesskiĭ and M. B. Smirnov, Associative rings satisfying the identity of Lie solvability, Vestsī Akadèmīī Navuk BSSR. Seryya Fīzīka-Matèmatychnykh Navuk 2 (1982), 15–20.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the second author was supported by NSERC of Canada under grant # RGPIN 418201.
Rights and permissions
About this article
Cite this article
Siciliano, S., Usefi, H. Lie structure of smash products. Isr. J. Math. 217, 93–110 (2017). https://doi.org/10.1007/s11856-017-1439-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1439-5