Symmetric Elements and Identities in Group Algebras

Let FG be the group ring of a group G over a field F of characteristic p ≥ 0. Let * be the natural involution, γ = ∑γ(g)g → γ* = ∑γ(g)g-1. Let us denote by <math display='block'> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mi>K</mi><mi>G</mi> </mrow> <mo>)</mo></mrow> </mrow> <mo>+</mo> </msup> <mo>=</mo><mrow><mo>{</mo> <mrow> <mi>&#x03B3;</mi><mo>&#x2208;</mo><mi>F</mi><mi>G</mi><mo>:</mo><mi>&#x03B3;</mi><mo>*</mo><mo>=</mo><mi>&#x03B3;</mi> </mrow> <mo>}</mo></mrow><mtext>and</mtext><msup> <mrow> <mrow><mo>(</mo> <mrow> <mi>K</mi><mi>G</mi> </mrow> <mo>)</mo></mrow> </mrow> <mo>&#x2212;</mo> </msup> <mo>=</mo><mrow><mo>{</mo> <mrow> <mi>&#x03B3;</mi><mo>&#x2208;</mo><mi>F</mi><mi>G</mi><mo>:</mo><mi>&#x03B3;</mi><mo>*</mo><mo>=</mo><mo>&#x2212;</mo><mi>&#x03B3;</mi> </mrow> <mo>}</mo></mrow><mo>,</mo> </mrow> </math> $${{\left( {KG} \right)}^{+}}=\left\{ {\gamma \in FG:\gamma *=\gamma } \right\}\text{and}{{\left( {KG} \right)}^{-}}=\left\{ {\gamma \in FG:\gamma *=-\gamma } \right\},$$ the sets of symmetric and skew symmetric elements respectively. We investigate whether certain identities on these and similar subsets control identities on the whole ring.