Free group algebras in the field of fractions of differential polynomial rings and enveloping algebras
Let \(K\) be a characteristic 0 division ring, let \(\delta\) be a nonzero derivation of \(K\), and let \(R=K[X,X^{-1};\delta]\) be the subring of the division ring of fractions \(K(X;\delta)\) generated by \(K\), \(X\) and \(X^{-1}\). If \(A\) is a nonempty subset of \(K\), then the main result here is a collection of rather technical, but verifiable, conditions which guarantee that the elements \(aX^{-1}\) with \(a\in A\) freely generate a \(\mathbb{Q}\)-subalgebra of \(R\). With this in hand, the paper goes on to obtain a number of especially nice consequences. For example, Theorem C asserts that if \(D\) is a characteristic 0 division ring with an uncountable center and if \(D\) has an element \(x\) with \(D\neq C_D(x)\oplus[x,D]\), then \(D\) contains a noncommutative free group algebra. Furthermore, suppose \(L\) is a characteristic 0 Lie algebra and let \(D(L)\) denote the division ring generated by the enveloping algebra \(U(L)\) under the embedding given by \textit{P.~M.~Cohn} [Proc. Lond. Math. Soc., III. Ser. 11, 511-530 (1961; Zbl 0104.03203)]. Using this notation, Theorem D asserts that if \(L\neq L'+C_L(x)\) for some \(x\in L\setminus L'\), then \(D(L)\) contains a noncommutative free group algebra.
scientific article; zbMATH DE number 1218664
scientific article