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Article: Primitive, almost primitive, test, and Δ-primitive elements of free algebras with the Nielsen-Schreier property
Title | Primitive, almost primitive, test, and Δ-primitive elements of free algebras with the Nielsen-Schreier property |
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Authors | |
Issue Date | 2000 |
Publisher | Academic Press. The Journal's web site is located at http://www.elsevier.com/locate/jalgebra |
Citation | Journal Of Algebra, 2000, v. 228 n. 2, p. 603-623 How to Cite? |
Abstract | We study generalized primitive elements of free algebras of finite ranks with the Nielsen-Schreier property and their automorphic orbits. A primitive element of a free algebra is an element of some free generating set of this algebra. Almost primitive elements are not primitive elements which are primitive in any proper subalgebra. Δ-primitive elements are elements whose partial derivatives generate the same one-sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate. We prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism. An algorithm to determine test elements of free algebras of rank two is described. A series of almost primitive elements is constructed and new examples of test elements are given. We prove that if the rank n of the free Lie algebra L is even, n=2m, then any Δ-primitive element of L is an automorphic image of the element w=[x1,x2]+···+[x2m-1,x2m], there are no Δ-primitive elements of L if n is odd, and the group of automorphisms of the algebra L acts transitively on the set of all Δ-primitive elements. © 2000 Academic Press. |
Persistent Identifier | http://hdl.handle.net/10722/156084 |
ISSN | 2015 Impact Factor: 0.66 2015 SCImago Journal Rankings: 1.165 |
ISI Accession Number ID | |
References |
DC Field | Value | Language |
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dc.contributor.author | Mikhalev, AA | en_US |
dc.contributor.author | Yu, JT | en_US |
dc.date.accessioned | 2012-08-08T08:40:20Z | - |
dc.date.available | 2012-08-08T08:40:20Z | - |
dc.date.issued | 2000 | en_US |
dc.identifier.citation | Journal Of Algebra, 2000, v. 228 n. 2, p. 603-623 | en_US |
dc.identifier.issn | 0021-8693 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/156084 | - |
dc.description.abstract | We study generalized primitive elements of free algebras of finite ranks with the Nielsen-Schreier property and their automorphic orbits. A primitive element of a free algebra is an element of some free generating set of this algebra. Almost primitive elements are not primitive elements which are primitive in any proper subalgebra. Δ-primitive elements are elements whose partial derivatives generate the same one-sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate. We prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism. An algorithm to determine test elements of free algebras of rank two is described. A series of almost primitive elements is constructed and new examples of test elements are given. We prove that if the rank n of the free Lie algebra L is even, n=2m, then any Δ-primitive element of L is an automorphic image of the element w=[x1,x2]+···+[x2m-1,x2m], there are no Δ-primitive elements of L if n is odd, and the group of automorphisms of the algebra L acts transitively on the set of all Δ-primitive elements. © 2000 Academic Press. | en_US |
dc.language | eng | en_US |
dc.publisher | Academic Press. The Journal's web site is located at http://www.elsevier.com/locate/jalgebra | en_US |
dc.relation.ispartof | Journal of Algebra | en_US |
dc.title | Primitive, almost primitive, test, and Δ-primitive elements of free algebras with the Nielsen-Schreier property | en_US |
dc.type | Article | en_US |
dc.identifier.email | Yu, JT:yujt@hku.hk | en_US |
dc.identifier.authority | Yu, JT=rp00834 | en_US |
dc.description.nature | link_to_subscribed_fulltext | en_US |
dc.identifier.doi | 10.1006/jabr.2000.8288 | en_US |
dc.identifier.scopus | eid_2-s2.0-0034658810 | en_US |
dc.identifier.hkuros | 53209 | - |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-0034658810&selection=ref&src=s&origin=recordpage | en_US |
dc.identifier.volume | 228 | en_US |
dc.identifier.issue | 2 | en_US |
dc.identifier.spage | 603 | en_US |
dc.identifier.epage | 623 | en_US |
dc.identifier.isi | WOS:000087678800014 | - |
dc.publisher.place | United States | en_US |
dc.identifier.scopusauthorid | Mikhalev, AA=7007020926 | en_US |
dc.identifier.scopusauthorid | Yu, JT=7405530208 | en_US |