The Inversion Height of the Free Field is Infinite

TL;DR

This paper proves that the inversion height of the embedding of a free algebra into its universal skew field of fractions is infinite, answering a long-standing question and extending known results about fields of fractions.

Contribution

It establishes the infinite inversion height for free algebra embeddings and extends the understanding of fields of fractions for crossed products involving free Lie algebras.

Findings

Inversion height of free algebra embedding is infinite.

The result confirms a question by Neumann from 1949.

Universal fields of fractions for certain crossed products are constructed by Cohn and Lichtman.

Abstract

Let X be a finite set with at least two elements, and let k be any commutative field. We prove that the inversion height of the embedding k<X> ---> D, where D denotes the universal (skew) field of fractions of the free algebra k<X>, is infinite. Therefore, if H denotes the free group on X, the inversion height of the embedding of the group algebra k[H] into the Malcev-Neumann series ring is also infinite. This answer in the affirmative a question posed by Neumann in 1949 [27, p. 215]. We also give an infinite family of examples of non-isomorphic fields of fractions of k<X> with infinite inversion height. We show that the universal field of fractions of a crossed product of a commutative field by the universal enveloping algebra of a free Lie algebra is a field of fractions constructed by Cohn (and later by Lichtman). This extends a result by A. Lichtman.