Lie Nilpotent Group Algebras: A Survey

Reetu Siwach

doi:https://doi.org/10.14445/22315373/IJMTT-V65I12P504

International Journal of Mathematics Trends and Technology

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Volume 65 | Issue 12 | Year 2019 | Article Id. IJMTT-V65I12P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I12P504

Lie Nilpotent Group Algebras: A Survey

Reetu Siwach

Abstract

Let R be any ring with unity. Then R can be treated as a Lie ring under the Lie multiplication [x'y] = xy-yx; x,y ε R. This ring is denoted by L(R) and is called associated Lie ring of R.

Keywords

Survey, Nilpotent, Algebras

References

[1] A. K. Bhandari and I. B. S. Passi, Lie nilpotency indices of group algebras, Bull. London Math. Soc. 24(1992)68 - 70.
[2] V. Bovdi, Modular group algebras with almost maximal Lie nilpotency in-dicesII, Sci. Math. Jpn. 65(2007)267 - 271.
[3] V. Bovdi, T. Juhasz, E. Spinelli, Modular group algebras with almost maximal Lie nilpotency indices, Algebr. Represent.Theory 9(3)(2006)259 - 266.
[4] A. A. Bovdi and I. I. Khripta, Generalized Lie nilpotent group rings, Math. USSR Sb.57(1)(1987)165 - 169.
[5] A. A. Bovdi and J. Kurdics, Lie properties of the group algebras and nilpotency class of the group of units. J. Algebra 212(1)(1999)28 - 64.
[6] V. Bovdi and E. Spinelli, Modular group algebras with maximal Lie nilpotency indices, Publ.Math.Debrecen. 65(1 - 2)(2004)243 - 252.
[7] V. Bovdi and J. B. Srivastava, Lie nilpotency indices of modular group algebras, Algebra Colloq.17(2010)17 - 26.
[8] H. Chandra and M. Sahai, Strongly Lie Nilpotent group algebras of index atmost 8 J.Algebra and its Appl.13(2014).
[9] I. B .S. Passi, Group Rings and their Augmentation Ideals, Lecture Notes in Mathematics, No.715(Springer-Verlag,Berlin,1979).
[10] I.B.S. Passi, D.S. Passman, S. K. Sehgal, Lie solvable group rings,Canad. J. Math. 25 (1973)748 - 757:
[11] R. Siwach, R. K. Sharma, M. Sahai, On the Lie nilpotency indices of modular group algebras, Beitr. Algebra Geom.58(2) (2017)355 - 367.
[12] R. Siwach, R. K. Sharma, M. Sahai, Modular group algebras with small Lie nilpotency indices,Asian Eur. J. Math 9(2) 1650080(2016).
[13] R. Siwach, R. K. Sharma, M. Sahai, Strongly Lie solvable group algebras of derived length 4,Beitr. Algebra Geom. 57(4) (2016)881 - 889.
[14] M. Sahai, R. Siwach, R. K. Sharma, On the Lie nilpotency indices of modular group algebras II,Asian Eur. J. Math 11(6) 1850087(2018).
[15] R. K. Sharma, R. Siwach, M. Sahai, Group algebras of Lie nilpotency index 12 and 13 ,Comm. Algebra 46(4) (2018)1428 - 1446.
[16] A. Shalev, Applications of dimension and Lie dimension subgroups to modular group algebras, in Proc.Amitsur Conf. in Ring Theory (1989); pp.84-95.
[17] A. Shalev, Lie dimension subgroups, Lie nilpotency indices and the exponent of the group of normalized units, J.London Math.Soc. 43(1991)23 - 36.
[18] A. Shalev, The nilpotency class of the unit group of a modular group algebra III, Arch. Math. 60(1993)136 - 145.
[19] R. K. Sharma and V. Bist, A note on Lie nilpotent group rings, Bull. Autral. Math. Soc. 45(1992)503 - 506.
[20] R. K. Sharma and J. B. Srivastava, Lie ideals in group rings, J. Pure Appl. Algebra. 63(1990)67 - 80.
[21] M. Sahai and B. Sharan, On Lie nilpotent modular group algebras, Comm. Algebra 46(3) 2018.
[22] I. B. S. Passi, D. S. Passman, and S. K. Sehgal. Lie solvable group rings, Canad. J. Math 25 (1973)748 - 757.
[23] M. Sahai. and B. Sharan, Group algebras with almost minimal Lie nilpotency index, Publ. Math. Debrecen 87(1-2)(2015)4755.
[24] M. Sahai. and B. Sharan, A note on Lie nilpotent group algebras ,J. Algebra and its Appl. - (2019) 12pages.
[25] A. Shalev, The nilpotency class of the unit group of a modular group algebra III, Arch. Math. 60(1993)136 - 145.
[26] V. Bovdi and E. Spinelli, Modular group algebras with maximal Lie nilpotency indices. Publ. Math. (Debr.) 65/12;(2004)243252.
[27] A.B. Konovalov, Calculations in group algebras using the GAP package LAG 3.0, Nicolaus Conference 2002, LDFM, RWTH Aachen, 2002, 191200.
[28] M. Sahai and B. Sharan, A note on the upper Lie nilpotency index of a group algebra ,Asian Eur. 12(1)(2020)11pages.
[29] N. Gupta and F. Levin, On Lie ideals of rings ,J.Algebra 81(1983)225-231.
[30] S. Bhatt, H. Chandra and M. Sahai, Group Algebras of Lie nilpotency index 14 ,Asian Eur. J. Math DOI : 10:1142/S1793557120500886.

Citation :

Reetu Siwach, "Lie Nilpotent Group Algebras: A Survey," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 12, pp. 36-43, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I12P504