The inversion height of the free field is infinite
Herbera i Espinal, Dolors (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Sánchez, Javier (University of São Paulo. Department of Mathematics)

Data:	2015
Resum:	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 kH 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 k(X) 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.
Ajuts:	Ministerio de Economía y Competitividad MTM2011-28992-C02-01 Generalitat de Catalunya 2009/SGR-1389
Nota:	Altres ajuts: ERDF UNAB10-4E-378 "A way to build Europe"
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Publicat a:	Selecta Mathematica New Series, Vol. 21, Num. 3 (July 2015) , p. 883-929, ISSN 1420-9020

DOI: 10.1007/s00029-014-0168-4

Postprint 47 p, 511.9 KB

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