Free symmetric pairs in the field of fractions of enveloping Lie algebras with involution

Abstract

Let [Formula: see text] be a division ring with center [Formula: see text], [Formula: see text], let [Formula: see text] be an involution of [Formula: see text], and let [Formula: see text] be the multiplicative group of [Formula: see text]. A pair [Formula: see text] is called free symmetric, if it is formed by symmetric elements, and it generates a free non-cyclic subgroup of [Formula: see text]. If [Formula: see text] is the enveloping algebra of the non-abelian nilpotent Lie [Formula: see text]-algebra [Formula: see text] over the field [Formula: see text] of characteristic [Formula: see text], and [Formula: see text] is a [Formula: see text]-involution of [Formula: see text] extended to the field of fractions [Formula: see text] of [Formula: see text], we show that [Formula: see text] contains free symmetric pairs. We also discuss the consequences of symmetric elements of a normal subgroup being torsion over the center.