Free algebras generated by symmetric elements inside division rings with involution

This is a joint work with Vitor O. Ferreira and Jairo Z. Gon ̧calves. For any Lie algebra L over a field, its universal enveloping algebra U (L) can be embedded in a division ring D (L) constructed by Cohn [?] (and simplified later by Lichtman [?]). If U (L) is an Ore domain, D (L) coincides with its ring of fractions. Consider now the principal involution of L, L → L, x 7→ − x. It is well known that the principal involution of L can be extended to an involution of U (L). It was proved by Cimpric, that this involution can be extended to D (L) [?]. For a large class of noncommutative Lie algebras L over a field of zero charac-teristic, we show that D (L) contains noncommutative free algebras generated by symmetric elements (with respect to the extension of the pri ncipal involution). This class contains all noncommutative Lie algebras over a field of zero characteristic such that U(L) is an Ore domain