The inversion height of the free field is infinite

Author(s)

Herbera, Dolors ^[1] ; Sanchez, Javier ^[2]

Affiliation

^[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona - Spain

^[2] Univ Sao Paulo, IME, Dept Math, BR-05314970 Sao Paulo, SP - Brazil

• Document type: Journal article
• Source: SELECTA MATHEMATICA-NEW SERIES; v. 21, n. 3, p. 883-929, JUL 2015.
• Web of Science Citations: 3

Abstract

Let be a finite set with at least two elements, and let be any commutative field. We prove that the inversion height of the embedding , where denotes the universal (skew) field of fractions of the free algebra , is infinite. Therefore, if denotes the free group on , the inversion height of the embedding of the group algebra into the Malcev-Neumann series ring is also infinite. This answers in the affirmative a question posed by Neumann (Trans Am Math Soc 66:202-252, 1949). We also give an infinite family of examples of non-isomorphic fields of fractions of with infinite inversion height. We show that the universal field of fractions of a crossed product of a 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. (AU)

• FAPESP's process: 09/50886-0 - Embedding group algebras and crossed products in division rings
• Grantee: JAVIER SANCHEZ SERDA
• Support Opportunities: Scholarships in Brazil - Post-Doctoral