Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:14:09.400Z Has data issue: false hasContentIssue false

Abelian Gradings on Upper Block Triangular Matrices

Published online by Cambridge University Press:  20 November 2018

Angela Valenti
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Universitá di Palermo, Palermo, Italy e-mail: avalenti@unipa.it
Mikhail Zaicev
Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia e-mail: zaicev@mech.math.msu.su
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $G$ be an arbitrary finite abelian group. We describe all possible $G$-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Bahturin, Y. A., Montgomery, S., and Zaicev, M. V., Generalized Lie solvability of associative algebras. In: Groups, Rings, Lie and Hopf Algebras. Math. Appl. 555, Kluwer, Dordrecht, 2003, pp. 123 Google Scholar
[2] Bahturin, Y. A., Sehgal, S. K., and Zaicev, M. V., Group gradings on associative algebras. J. Algebra 241(2001), no. 2, 677698. doi:10.1006/jabr.2000.8643Google Scholar
[3] Bahturin, Y. A., Shestakov, I., and Zaicev, M. V., Gradings on simple Jordan algebras and Lie algebras. J. Algebra 283(2005), 849868. doi:10.1016/j.jalgebra.2004.10.007Google Scholar
[4] Bahturin, Y. A. and Zaicev, M. V., Group gradings on matrix algebras. Dedicated to Robert V. Moody. Canad. Math. Bull. 45 (2002), no. 4, 499508. doi:10.4153/CMB-2002-051-xGoogle Scholar
[5] Bahturin, Y. A. and Zaicev, M. V., Identities of graded algebras and codimension growth. Trans. Amer. Math. Soc. 356(2004), no. 10, 39393950. doi:10.1090/S0002-9947-04-03426-9Google Scholar
[6] Di Vincenzo, O. M., Koshlukov, P., and Valenti, A., Gradings on the algebra of upper triangular matrices and their graded identities. J. Algebra 275(2004), no. 2, 550556. doi:10.1016/j.jalgebra.2003.08.004Google Scholar
[7] Giambruno, A. and Zaicev, M., Polynomial Identities and Asymptotic Methods. Mathematical Surveys and Monographs 122, American Mathematical Society, Providence, RI, 2005.Google Scholar
[8] Jacobson, N., The Theory of Rings American Mathematical Society Math. Surveys 2, American Mathematical Society, New York, 1943.Google Scholar
[9] Kantor, I. L., Some generalizations of Jordan algebras. Trudy Sem. Vektor. Tenzor. Anal. 16(1972), 407499.Google Scholar
[10] Sehgal, S. K., Topics in Group Rings. Monographs and Textbooks in Pure and Applied Math. 50, Marcel Dekker, New York, 1978.Google Scholar
[11] Sehgal, S. K. and Zaicev, M. V., Graded identities and induced gradings on group algebras. In: Groups, Rings, Lie and Hopf Algebras, Mathematics Appl. 555, Kluwer Acad. Public. 2003, pp. 211219.Google Scholar
[12] Smirnov, O. N., Simple associative algebras with finite Z -grading. J. Algebra 196(1997), 171184. doi:10.1006/jabr.1997.7087Google Scholar
[13] Smirnov, O. N., Finite Z -grading of Lie algebras and symplectic involution. J. Algebra 218(1999), no. 1, 246275. doi:10.1006/jabr.1999.7880Google Scholar
[14] Valenti, A. and Zaicev, M. V., Abelian gradings on upper-triangular matrices. Arch. Math. 80(2003), no. 1, 1217.Google Scholar
[15] Valenti, A. and Zaicev, M. V., Group gradings on upper triangular matrices. Arch. Math. 89(2007), no. 1, 3340.Google Scholar
[16] Zaĭtsev, M. V. and Segal, S. K., Finite gradings of simple Artinian rings. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 77(2001) no. 3, 21–24; translation in Moscow Univ. Math. Bull. 56(2001), no. 3, 2124.Google Scholar