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Involutions and Anticommutativity in Group Rings

Published online by Cambridge University Press:  20 November 2018

Edgar G. Goodaire
Affiliation:
Memorial University of Newfoundland, St. John's, NF, A1C 5S7 e-mail: [email protected]
César Polcino Milies
Affiliation:
Memorial University of Newfoundland, St. John's, NF, A1C 5S7 e-mail: [email protected]
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Abstract

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Let $g\,\mapsto \,{{g}^{*}}$ denote an involution on a group $G$. For any (commutative, associative) ring $R$ (with 1), $*$ extends linearly to an involution of the group ring $RG$. An element $\alpha \,\in \,RG$ is symmetric if ${{\alpha }^{*}}\,=\,\alpha $ and skew-symmetric if ${{\alpha }^{*}}\,=\,-\alpha $. The skew-symmetric elements are closed under the Lie bracket, $[\alpha ,\,\beta ]\,=\,\alpha \beta \,-\,\beta \alpha $. In this paper, we investigate when this set is also closed under the ring product in $RG$. The symmetric elements are closed under the Jordan product, $\alpha \,o\,\beta \,=\,\alpha \beta \,+\beta \alpha $. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Chein, O. and Goodaire, E. G., Code loops are RA2 loops. J. Algebra 130(1990), no. 2, 385387. http://dx.doi.org/10.1016/0021-8693(90)90088-6 Google Scholar
[2] Broche Cristo, O., Commutativity of symmetric elements in group rings. J. Group Theory 9(2006), no. 5, 673683. http://dx.doi.org/10.1515/JGT.2006.043 Google Scholar
[3] Broche Cristo, O., Jespers, E., C. Polcino Milies, and M. Ruiz Marĭn, Antisymmetric elements in group rings. II. J. Algebra Appl. 8(2009), no. 1, 115127. http://dx.doi.org/10.1142/S0219498809003254 Google Scholar
[4] Broche Cristo, O. and Ruiz Marĭn, M., Lie identities in symmetric elements in group rings: A survey. In: Groups, Rings and Group Rings. Lect Notes Pure Appl. Math. 248. Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 4355. Google Scholar
[5] Broche Cristo, O. and Polcino Milies, C., Commutativity of skew symmetric elements in group rings. Proc. Edinb. Math. Soc. 50(2007), no. 1, 3747. http://dx.doi.org/10.1017/S0013091504000896 Google Scholar
[6] Giambruno, A. and Polcino Milies, C., Unitary units and skew elements in group algebras. Manuscripta Math. 111(2003), no. 2, 195209. http://dx.doi.org/10.1007/s00229-003-0365-5 Google Scholar
[7] Giambruno, A. and Sehgal, S. K., Lie nilpotence of group rings. Comm. Algebra 21(1993), no. 11, 42534261. http://dx.doi.org/10.1080/00927879308824797 Google Scholar
[8] Giambruno, A., Polcino Milies, C., and Sehgal, S. K., Group algebras of torsion groups and Lie nilpotence. J. Group Theory. 13(2009), no. 2, 221231. http://dx.doi.org/10.1515/JGT.2009.048 Google Scholar
[9] Giambruno, A., C. Polcino Milies, , and Sehgal, S. K., Group identities on symmetric units. J. Algebra 322(2009), no. 8, 28012815. http://dx.doi.org/10.1016/j.jalgebra.2009.06.025 Google Scholar
[10] Giambruno, A., Polcino Milies, C., and Sehgal, S. K., Lie properties of symmetric elements in group rings. J. Algebra 321(2009), no. 3, 890902. http://dx.doi.org/10.1016/j.jalgebra.2008.09.041 Google Scholar
[11] Giambruno, A., Sehgal, S. K., and Valenti, A., Symmetric units and group identities. Manuscripta Math. 96(1998), no. 4, 443461. http://dx.doi.org/10.1007/s002290050076 Google Scholar
[12] Gonçalves, J. Z. and Passman, D. S., Involutions and free pairs of bicyclic units in integral group rings. J. Group Theory 13(2010), no. 5, 721742. http://dx.doi.org/10.1515/JGT.2010.019 Google Scholar
[13] Gonçalves, J. Z. and Passman, D. S., Unitary units in group algebras. Israel J. Math. 125(2001), 131155. http://dx.doi.org/10.1007/BF02773378 Google Scholar
[14] Goodaire, E. G., Groups embeddable in alternative loop rings. In: Contributions to General Algebra 7. Hölder-Pichler-Tempsky, Vienna, 1991, pp. 169176. Google Scholar
[15] Goodaire, E. G., Jespers, E., and C. Polcino Milies, Alternative loop rings. North-Holland Mathematics Studies 184. North-Holland Publishing, Amsterdam, 1996.Google Scholar
[16] Jespers, E. and Ruiz Marĭn, M., Antisymmetric elements in group rings. J. Algebra Appl. 4(2005), no. 4, 341353. http://dx.doi.org/10.1142/S0219498805001228 Google Scholar
[17] Jespers, E. and Ruiz Marĭn, M., On symmetric elements and symmetric units in group rings. Comm. Algebra 34(2006), no. 2, 727736. http://dx.doi.org/10.1080/00927870500388018 Google Scholar
[18] Lee, G. T., Group identities on units and symmetric units of group rings. Algebra and Applications 12. Springer-Verlag, London, 2010.Google Scholar
[19] Lee, G. T., Sehgal, S. K., and Spinelli, E., Group algebras whose symmetric and skew elements are Lie solvable. Forum Math. 21(2009), no. 4, 661671. http://dx.doi.org/10.1515/FORUM.2009.033 Google Scholar
[20] Lee, G. T., Sehgal, S. K., and Spinelli, E., Lie properties of symmetric elements in group rings. II. J. Pure Appl. Algebra 213(2009), no. 6, 11731178. http://dx.doi.org/10.1016/j.jpaa.2008.11.027 Google Scholar
[21] Sehgal, S. K. and Valenti, A., Group algebras with symmetric units satisfying a group identity. Manuscripta Math. 119(2006), no. 2, 243254. http://dx.doi.org/10.1007/s00229-005-0610-1 Google Scholar