Forum Math. 2017; aop Research Article Antonio Giambruno*, Cesar Polcino Milies and Sudarshan K. Sehgal Star-group identities on units of group algebras: The non-torsion case DOI: 10.1515/forum-2016-0266 Received December 27, 2016; revised May 3, 2017 Abstract: Let G be a group, F a őeld and FG the corresponding group algebra. We consider an involution on FG which is the linear extension of an involution of G, e.g., g ∗ = g −1 for g ∈ G. This paper is focused on the characterization of a non-torsion group G provided the group of units U(FG) satisőes a ∗-group identity. The torsion case was studied in [7], and when ∗ is the classical involution, this problem was solved in the case of symmetric units in [21]. Keywords: Group algebra, involution, unit, group identity MSC 2010: Primary 16U60; secondary 16W10 || Communicated by: Manfred Droste 1 Introduction Let FG be the group algebra of a group G over a őeld F and let U(FG) denote its group of units. U(FG) has been extensively studied in the past (see for instance [10, 16, 20]), and it turns out that, even in the case of őnite groups, it is quite large and difcult to study. In an attempt to tie the group of units to the structure of the algebra, in the 80s Hartley conjectured that for torsion groups, a group identity on U(FG) would force a polynomial identity on FG. This conjecture was proved in the 90s in [4, 8, 13, 15], and it turns out that one can actually obtain a classiőcation of torsion groups G such that the group of units U(FG) satisőes a group identity. These results were the starting point for the development of a theory of group identities on U(FG): on the one hand, the investigation was carried over to non-torsion groups [8], and on the other hand, group identities on signiőcant units of U(FG) were studied, such as symmetric units or unitary units [3, 5, 12, 21]. In this last setting, since a group algebra FG is always endowed with an involution, one can consider involutions ∗ obtained as a linear extension of an involution of the group G (see [2, 9]). One such involution is for instance the so-called classical involution deőned on G by g ∗ = g −1 (notice that any other involution is the composition of an automorphism of G of order 1 or 2 with the inverse map). In this case, a natural extension of Hartley’s conjecture and its further development is to consider group identities on symmetric units, i.e., words of the free group evaluating onto the identity when computed on symmetric units of FG. *Corresponding author: Antonio Giambruno: Dipartimento di Matematica ed Informatica, Università di Palermo, Via Archiraő 34, 90123 Palermo, Italy, e-mail: antonio.giambruno@unipa.it. http://orcid.org/0000-0002-3422-2539 Cesar Polcino Milies: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, CEP-05315-970, São Paulo, Brazil, e-mail: polcino@ime.usp.br Sudarshan K. Sehgal: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada, e-mail: s.sehgal@ualberta.ca Brought to you by | Cornell University Library Authenticated Download Date | 7/9/17 9:44 AM