{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T05:40:42Z","timestamp":1687930842819},"reference-count":7,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2013,2]]},"abstract":"<jats:p>Let L be a free Lie algebra of finite rank over a field K and let L<jats:sub>n<\/jats:sub>denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [L<jats:sub>m<\/jats:sub>, L<jats:sub>n<\/jats:sub>] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [L<jats:sub>m<\/jats:sub>, L<jats:sub>n<\/jats:sub>, L<jats:sub>k<\/jats:sub>] = [[L<jats:sub>m<\/jats:sub>, L<jats:sub>n<\/jats:sub>], L<jats:sub>k<\/jats:sub>]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L<jats:sub>2<\/jats:sub>, L<jats:sub>2<\/jats:sub>, L<jats:sub>1<\/jats:sub>] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [L<jats:sub>m<\/jats:sub>, L<jats:sub>n<\/jats:sub>, L<jats:sub>k<\/jats:sub>]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [L<jats:sub>m<\/jats:sub>, L<jats:sub>n<\/jats:sub>, L<jats:sub>k<\/jats:sub>] in terms of Witt's dimension function.<\/jats:p>","DOI":"10.1142\/s0218196713500069","type":"journal-article","created":{"date-parts":[[2012,12,26]],"date-time":"2012-12-26T01:21:35Z","timestamp":1356484895000},"page":"205-213","source":"Crossref","is-referenced-by-count":1,"title":["ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS"],"prefix":"10.1142","volume":"23","author":[{"given":"NIL","family":"MANSURO\u01e6LU","sequence":"first","affiliation":[{"name":"School of Mathematics University of Manchester, Alan Turing Building, Manchester, M13 9PL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RALPH","family":"ST\u00d6HR","sequence":"additional","affiliation":[{"name":"School of Mathematics University of Manchester, Alan Turing Building, Manchester, M13 9PL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2013,3,3]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700022606"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1080\/00927877608822105"},{"key":"rf4","volume-title":"Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations","author":"Magnus W.","year":"1976"},{"key":"rf5","first-page":"441","volume":"33","author":"Shirshov A. I.","year":"1953","journal-title":"Mat. Sb."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196709005287"},{"key":"rf8","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1515\/crll.1937.177.152","volume":"177","author":"Witt E.","year":"1937","journal-title":"J. Reine Angew. Math."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01166568"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196713500069","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T05:06:10Z","timestamp":1687928770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196713500069"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2]]},"references-count":7,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2013,3,3]]},"published-print":{"date-parts":[[2013,2]]}},"alternative-id":["10.1142\/S0218196713500069"],"URL":"https:\/\/doi.org\/10.1142\/s0218196713500069","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2]]}}}