Free Group Algebras in Division Rings with Valuation II

Abstract

We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If L is a Lie algebra, we denote by U (L) its universal enveloping algebra. P. M. Cohn constructed a division ring D L that contains U (L). We denote by D(L) the division subring of D L generated by U (L).Let k be a field of characteristic zero and L be a nonabelian Lie k-algebra. If either L is residually nilpotent or U (L) is an Ore domain, we show that D(L) contains (noncommutative) free group algebras. In those same cases, if L is equipped with an involution, we are able to prove that the free group algebra in D(L) can be chosen generated by symmetric elements in most cases.Let G be a nonabelian residually torsion-free nilpotent group and k(G) be the division subring of the Malcev-Neumann series ring generated by the group algebra k[G]. If G is equipped with an involution, we show that k(G) contains a (noncommutative) free group algebra generated by symmetric elements.

Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

“…For a proof, the same argument used in the proof of Theorem 3.3 shows that if A and B are symmetric elements in k(G/N ) generating a free group algebra, then X and Y will be symmetric elements in D G generating a free group algebra. That k(G/N ) contains such a pair follows from [20]. Similarly, one can use this version of Theorem 3.3 to prove that if k has characteristic zero and G is a nonabelian residually torsion-free nilpotent group with an involution * , then the Malcev-Neumann division ring of fractions k(G) of k[G] contains a free group k-algebra generated by symmetric elements with respect to the k-involution on k(G) induced by * .…”