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

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 → −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.

Free Group Algebras in Division Rings with Valuation II

“…In this section we find different free (group) k-subalgebras in D(H), the Ore ring of fraction of the universal enveloping algebra U (H) of H. For that, our main tool is the result by G. Cauchon [4,Théorème]. The technique to obtain suitable free algebras from the paper of Cauchon was developed in [10,Section 3]. In some cases, we then use Theorem 2.7 to obtain free group algebras in D(H).…”

“…In this subsection, we want to obtain free group algebras in D(L), where L is a (generalization of) a residually nilpotent Lie algebra, from the ones obtained in D(H), where H is the Heisenberg Lie algebra. The technique we will use is from [10,Section 4].…”

“…Let k be a field of characteristic zero. The general strategy to obtain Theorems 1.1 and 1.2 goes back to Lichtman [25], and it was also used in [10]. Roughly speaking, one has to obtain free (group) algebras in the division ring D(H), where H is the Lie k-algebra H = x, y : [y, [y, x]] = [x, [y, x]] = 0 .…”

“…We have improved and somehow clarified this strategy in order to obtain the two first theorems above. Then Theorem 1.3 is obtained from the previous results using the filtered methods from [39] and a technique from [10].…”

“…Let L be a nonabelian residually nilpotent Lie k-algebra over a field of characteristic zero k. The main aim of Section 5 is to obtain free (group) algebras in the division ring D(L) from the ones obtained in the previous section. It is done by a method involving series that was developed in [10]. Although technical, the argument is quite natural.…”