Group algebras with symmetric units satisfying a Laurent polynomial identity (Q6635310)

This paper studies group algebras \(FG\), where \(F\) is an infinite field of characteristic different from \(2\), and \(G\) is a group. The main focus is on the symmetric units \N\[\N\mathcal U^+(FG) = \{\alpha\in \mathcal U(FG) |\alpha^* = \alpha\},\N\]\Nwhere \(*\) is the natural involution on \(FG\). The background and motivation come from previous work on group identities satisfied by the unit group \(\mathcal U(FG)\) and the symmetric units \(\mathcal U^+(FG)\). The author defines and studies the notion of ``normalized Laurent polynomial identities'' on \(\mathcal U^+(FG)\), which is a generalization of the well-known ``group identities'' on \(\mathcal U^+(FG)\).\N\NThe main results are:\N\N1. If \(\mathcal U^+(FG)\) satisfies a normalized Laurent polynomial identity, then \(FG\) satisfies a polynomial identity. This generalizes Hartley's conjecture, which states that if \(\mathcal U(FG)\) satisfies a group identity, then \(FG\) satisfies a polynomial identity.\N\N2. For torsion groups \(G\), \(\mathcal U^+(FG)\) satisfies a normalized Laurent polynomial identity if and only if \(G\) is abelian or a Hamiltonian 2-group, if and only if \(\mathcal U^+(FG)\) is abelian.\N\N3. For non-torsion groups \(G\), if \(\mathcal U^+(FG)\) satisfies a normalized Laurent polynomial identity, then the set of torsion elements \(T\) forms an abelian normal subgroup, and every idempotent in the group algebra \(F(T)\) is central in \(FG\).\N\NThe key methods use include utilizing the properties of the normalized Laurent polynomials, techniques from the structure theory of group algebras, and detailed analysis of the structure of the symmetric units \(\mathcal U^+(FG)\) under various assumptions on the group \(G\).\N\NThe paper provides a comprehensive understanding of when group algebras \(FG\) have symmetric units \(\mathcal U^+(FG)\) satisfying a normalized Laurent polynomial identity, and how this relates to the polynomial identity problem for \(FG\). The results unify and generalize previous work in this direction.