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Article: Degree estimate for commutators

TitleDegree estimate for commutators
Authors
KeywordsAutomorphisms
Commutators
Coordinates
Degree Estimate
Equation [Um, S] = [Un, R]
Free Associative Algebras
Jacobian
Jacobian Conjecture
K [U]-Bimodules
Polynomial Algebras
Tame Automorphisms
Wild Automorphisms
Issue Date2009
PublisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jalgebra
Citation
Journal Of Algebra, 2009, v. 322 n. 7, p. 2321-2334 How to Cite?
AbstractLet K 〈 X 〉 be a finite generated free associative algebra over a field K of characteristic zero and let f, g ∈ K 〈 X 〉 generate their centralizer respectively. Assume that the f and g are algebraically independent, the leading homogeneous components of f and g are algebraically dependent and deg (f) {does not divide} deg (g), deg (g) {does not divide} deg (f). In this article, we construct a counterexample to a conjecture of Jie-Tai Yu that deg ([f, g]) > min {deg (f), deg (g)}, which is closely related to the study of the structure of the automorphism group of K 〈 X 〉. We also obtain a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev-Neumann formal power series. In view of the counterexamples we formulate two open problems concerning degree estimate for commutators in view of the study of the structure of the automorphism group of K 〈 X 〉. The counterexamples in this article are constructed by applying the description of the free algebra K 〈 X 〉 considered as a bimodule of K [u] where u is a monomial which is not a power of another monomial and the solution the equation [um, s] = [un, r] with unknowns r, s ∈ K 〈 X 〉. The newly discovered description and the solution of the equation in this article are closely related to the combinatorial and computational aspects of free associative algebras, hence have their own independent interests. © 2009 Elsevier Inc. All rights reserved.
Persistent Identifierhttp://hdl.handle.net/10722/156242
ISSN
2015 Impact Factor: 0.66
2015 SCImago Journal Rankings: 1.165
ISI Accession Number ID
Funding AgencyGrant Number
Bulgarian National Science FundMI-1503/2005
RGC-CERG
Funding Information:

Partially supported by an RGC-CERG grant.

References

 

DC FieldValueLanguage
dc.contributor.authorDrensky, Ven_US
dc.contributor.authorYu, JTen_US
dc.date.accessioned2012-08-08T08:40:59Z-
dc.date.available2012-08-08T08:40:59Z-
dc.date.issued2009en_US
dc.identifier.citationJournal Of Algebra, 2009, v. 322 n. 7, p. 2321-2334en_US
dc.identifier.issn0021-8693en_US
dc.identifier.urihttp://hdl.handle.net/10722/156242-
dc.description.abstractLet K 〈 X 〉 be a finite generated free associative algebra over a field K of characteristic zero and let f, g ∈ K 〈 X 〉 generate their centralizer respectively. Assume that the f and g are algebraically independent, the leading homogeneous components of f and g are algebraically dependent and deg (f) {does not divide} deg (g), deg (g) {does not divide} deg (f). In this article, we construct a counterexample to a conjecture of Jie-Tai Yu that deg ([f, g]) > min {deg (f), deg (g)}, which is closely related to the study of the structure of the automorphism group of K 〈 X 〉. We also obtain a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev-Neumann formal power series. In view of the counterexamples we formulate two open problems concerning degree estimate for commutators in view of the study of the structure of the automorphism group of K 〈 X 〉. The counterexamples in this article are constructed by applying the description of the free algebra K 〈 X 〉 considered as a bimodule of K [u] where u is a monomial which is not a power of another monomial and the solution the equation [um, s] = [un, r] with unknowns r, s ∈ K 〈 X 〉. The newly discovered description and the solution of the equation in this article are closely related to the combinatorial and computational aspects of free associative algebras, hence have their own independent interests. © 2009 Elsevier Inc. All rights reserved.en_US
dc.languageengen_US
dc.publisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jalgebraen_US
dc.relation.ispartofJournal of Algebraen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectAutomorphismsen_US
dc.subjectCommutatorsen_US
dc.subjectCoordinatesen_US
dc.subjectDegree Estimateen_US
dc.subjectEquation [Um, S] = [Un, R]en_US
dc.subjectFree Associative Algebrasen_US
dc.subjectJacobianen_US
dc.subjectJacobian Conjectureen_US
dc.subjectK [U]-Bimodulesen_US
dc.subjectPolynomial Algebrasen_US
dc.subjectTame Automorphismsen_US
dc.subjectWild Automorphismsen_US
dc.titleDegree estimate for commutatorsen_US
dc.typeArticleen_US
dc.identifier.emailYu, JT:yujt@hku.hken_US
dc.identifier.authorityYu, JT=rp00834en_US
dc.description.naturepostprinten_US
dc.identifier.doi10.1016/j.jalgebra.2009.07.018en_US
dc.identifier.scopuseid_2-s2.0-68349141527en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-68349141527&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume322en_US
dc.identifier.issue7en_US
dc.identifier.spage2321en_US
dc.identifier.epage2334en_US
dc.identifier.eissn1090-266X-
dc.identifier.isiWOS:000272261400005-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridDrensky, V=6603826254en_US
dc.identifier.scopusauthoridYu, JT=7405530208en_US
dc.identifier.citeulike5458243-

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