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Free Group Algebras in Division Rings with Valuation II
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 6 / December 2020
- Published online by Cambridge University Press:
- 05 July 2019, pp. 1463-1504
- Print publication:
- December 2020
-
- Article
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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 
$\mathfrak{D}_{L}$ that contains 
$U(L)$. We denote by 
$\mathfrak{D}(L)$ the division subring of 
$\mathfrak{D}_{L}$ generated by 
$U(L)$.Let
$k$ be a field of characteristic zero, and let 
$L$ be a nonabelian Lie 
$k$-algebra. If either 
$L$ is residually nilpotent or 
$U(L)$ is an Ore domain, we show that 
$\mathfrak{D}(L)$ 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 
$\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let
$G$ be a nonabelian residually torsion-free nilpotent group, and let 
$k(G)$ 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.
