Skew-symmetric elements of rational group algebras
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Skew-symmetric elements of rational group algebras Dishari Chaudhuri1 Received: 3 June 2019 / Accepted: 25 March 2020 © The Managing Editors 2020
Abstract Let RG be the group ring of a finite group G over a commutative ring R with 1. An element x in RG is said to be skew-symmetric with respect to an involution σ of RG if σ (x) = −x. A structure theorem for the Lie algebra of skew-symmetric elements of F G is given where F is an algebraic extension of Q which generalizes some previously known results in this direction. Keywords Group rings · Involution · Skew-symmetric elements · Lie ring · Central simple algebras Mathematics Subject Classification Primary 16S34; Secondary 16W10
1 Introduction Let RG be the group ring of a finite group G over a commutative ring R with 1 and let σ be an involution of RG. That is, σ : RG → RG is such that for x, y ∈ RG, σ satisfies the following conditions: (i) σ (x + y) = σ (x) + σ (y), (ii) σ (x y) = σ (y)σ (x) and − (iii) σ 2 (x) = x. Let RG + σ = {γ ∈ RG | σ (γ ) = γ } and RG σ = {γ ∈ RG | σ (γ ) = −γ } be the set of symmetric and skew-symmetric elements of RG respectively with respect to the involution σ . If σ is an R-linear extension of an involution of G, then RG − σ is generated by {g − σ (g) | g ∈ G\G σ } ∪ {rg | r ∈ R2 , g ∈ G σ } as an
The author was supported by DAE (Government of India) and National Board for Higher Mathematics with reference number 2/40(16)/2016/ R&D-II/5766 during this project. The author would like to thank IISER Mohali for providing good research facilities when this project was carried out. The author is very grateful to Abhay Soman for many insightful discussions. Prof. I.B.S. Passi deserves a special mention for kindling an interest of the author in the topic of the paper. Finally the author would like to thank the unknown referee whose valuable comments and suggestions helped a great deal in improving the exposition of this article.
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Dishari Chaudhuri [email protected]; [email protected] Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Sector-81, Knowledge City, S.A.S. Nagar, Mohali, Punjab 140306, India
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Beitr Algebra Geom
R-module, where G σ = {g ∈ G | σ (g) = g} and R2 = {r ∈ R | 2r = 0}. Note that RG − σ may not be a subring of RG in general. Now, RG may be viewed as a Lie ring with the help of the bracket operation [x, y] = x y − yx for x, y ∈ RG. Then, RG − σ becomes a Lie subring of RG. There are some strong relations between the structure of RG − σ and RG as Lie rings which have been studied by a few authors (for example, Amitsur 1969; Zaleskii and Smirnov 1981; Giambruno and Sehgal 1993). Also there are some close relations between polynomial identities satisfied by the unitary units of RG and the Lie algebra RG − σ (Giambruno and Milies 2003). The question of when is commutative has been completely answered in terms of the group elements RG − σ of G (Jespers and Ruiz Marín 2005, 2006; Cristo and Milies 2007; Cristo et al. 2009). Lie properties of RG − σ has been st
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