Elgendy, Hader A.; Bremner, Murray R. Universal associative envelopes of \((n+1)\)-dimensional \(n\)-Lie algebras. (English) Zbl 1260.17005 Commun. Algebra 40, No. 5, 1827-1842 (2012). Summary: For \(n\) even, we prove Pozhidaev’s conjecture on the existence of associative enveloping algebras for simple \(n\)-Lie (Filippov) algebras. More generally, for \(n\) even and any \((n+1)\)-dimensional \(n\)-Lie algebra \(L\), we construct a universal associative enveloping algebra \(U(L)\) and show that the natural map \(L\to U(L)\) is injective. We use noncommutative Gröbner bases to present \(U(L)\) as a quotient of the free associative algebra on a basis of \(L\) and to obtain a monomial basis of \(U(L)\). In the last section, we provide computational evidence that the construction of \(U(L)\) is much more difficult for \(n\) odd. Cited in 5 Documents MSC: 17A42 Other \(n\)-ary compositions \((n \ge 3)\) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 17B35 Universal enveloping (super)algebras Keywords:Pozhidaev’s conjecture; free associative algebras; n-Lie algebras; Filippov algebras; noncommutative Gröbner bases; representation theory; universal enveloping algebras PDFBibTeX XMLCite \textit{H. A. Elgendy} and \textit{M. R. Bremner}, Commun. Algebra 40, No. 5, 1827--1842 (2012; Zbl 1260.17005) Full Text: DOI arXiv References: [1] Bagger J., Phys. Rev. D 75 (4) pp 045020– (2007) · doi:10.1103/PhysRevD.75.045020 [2] Bai R., J. Phys. A 42 (3) pp 035207– (2009) · Zbl 1157.17002 · doi:10.1088/1751-8113/42/3/035207 [3] Bergman G. M., Adv. in Math. 29 (2) pp 178– (1978) · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5 [4] Casas J. M., J. Symbolic Comput. 42 (11) pp 1052– (2007) · Zbl 1131.17001 · doi:10.1016/j.jsc.2007.05.003 [5] de Azcárraga J. A., J. Phys. A: Math. Theor. 43 pp 293001– (2001) · Zbl 1202.81187 · doi:10.1088/1751-8113/43/29/293001 [6] de Graaf W. A., Lie Algebras: Theory and Algorithms (2000) [7] Filippov V. T., Sibirsk. Mat. Zh. 26 (6) pp 126– (1985) [8] Gustavsson A., Nuclear Phys. B 807 (1) pp 315– (2009) · Zbl 1192.81222 · doi:10.1016/j.nuclphysb.2008.09.003 [9] Humphreys J. E., Introduction to Lie Algebras and Representation Theory (1972) · Zbl 0254.17004 · doi:10.1007/978-1-4612-6398-2 [10] Insua M. A., J. Symbolic Comput. 44 (5) pp 517– (2009) · Zbl 1163.17004 · doi:10.1016/j.jsc.2007.07.020 [11] Ling , W. ( 1993 ).On the Structure ofn-Lie Algebras.Dissertation, University-GHS-Siegen, p. 61 . [12] Pojidaev A. P., Comm. Algebra 31 (2) pp 883– (2003) · Zbl 1025.17002 · doi:10.1081/AGB-120017349 [13] Bremner , M. R. ( 1998 ). Identities for the ternary commutator.J. Algebra206(2): 615–623 . · Zbl 0913.17001 [14] Bremner M. R., Linear Algebra Appl. 433 (8) pp 1686– (2010) · Zbl 1217.17003 · doi:10.1016/j.laa.2010.06.014 [15] Bremner , M. R. , Hentzel , I. R. ( 2006 ). Alternating triple systems with simple Lie algebras of derivations.Non-Associative Algebra and Its Applications.Lect. Notes Pure Appl. Math., 246. Boca Raton: Chapman & Hall/CRC, pp. 55–82 . · Zbl 1330.17006 [16] Cantarini N., Comm. Math. Phys. 298 (3) pp 833– (2010) · Zbl 1232.17008 · doi:10.1007/s00220-010-1049-0 [17] Curtright T., Phys. Lett. B 675 (3) pp 387– (2009) · doi:10.1016/j.physletb.2009.04.019 [18] Curtright T., J. Phys. A 42 (46) pp 462001– (2009) · Zbl 1178.81219 · doi:10.1088/1751-8113/42/46/462001 [19] Devchand C., J. Phys. A 42 (47) pp 475209– (2009) · Zbl 1230.17002 · doi:10.1088/1751-8113/42/47/475209 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.