Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

TL;DR

This paper demonstrates that for many noncommutative Lie algebras over characteristic zero fields, their division rings contain free algebras generated by symmetric elements related to the principal involution.

Contribution

It extends the understanding of free algebra existence within division rings generated by enveloping algebras of Lie algebras, especially for those with Ore domain properties.

Findings

Division rings contain noncommutative free algebras generated by symmetric elements.

Applicable to a large class of noncommutative Lie algebras in characteristic zero.

Includes all Lie algebras with enveloping algebras that are Ore domains.

Abstract

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 Lichtman. If U(L) is an Ore domain, D(L) coincides with its ring of fractions. It is well known that the principal involution of L, $x\mapsto -x$, can be extended to an involution of U(L), and Cimpric has proved that this involution can be extended to one on D(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that D(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.