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On upper modular subalgebras of a Lie algebra.

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On upper modular subalgebras of a Lie algebra. / Bowman, Kevin; Towers, David A.; Varea, Vicente R.
In: Proceedings of the Edinburgh Mathematical Society, Vol. 47, No. 2, 01.06.2004, p. 325-337.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Bowman, K, Towers, DA & Varea, VR 2004, 'On upper modular subalgebras of a Lie algebra.', Proceedings of the Edinburgh Mathematical Society, vol. 47, no. 2, pp. 325-337. https://doi.org/10.1017/S0013091503000051

APA

Bowman, K., Towers, D. A., & Varea, V. R. (2004). On upper modular subalgebras of a Lie algebra. Proceedings of the Edinburgh Mathematical Society, 47(2), 325-337. https://doi.org/10.1017/S0013091503000051

Vancouver

Bowman K, Towers DA, Varea VR. On upper modular subalgebras of a Lie algebra. Proceedings of the Edinburgh Mathematical Society. 2004 Jun 1;47(2):325-337. doi: 10.1017/S0013091503000051

Author

Bowman, Kevin ; Towers, David A. ; Varea, Vicente R. / On upper modular subalgebras of a Lie algebra. In: Proceedings of the Edinburgh Mathematical Society. 2004 ; Vol. 47, No. 2. pp. 325-337.

Bibtex

@article{bbc0038205ca44909f86a34708504ce4,
title = "On upper modular subalgebras of a Lie algebra.",
abstract = "This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper modular atom which is not an ideal. Finally it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a μ-algebra.",
keywords = "Keywords, Lie algebra, subalgebra lattice, upper modular",
author = "Kevin Bowman and Towers, {David A.} and Varea, {Vicente R.}",
note = "RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics http://journals.cambridge.org/action/displayJournal?jid=UHY The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 47 (2), pp 325-337 2004, {\textcopyright} 2004 Cambridge University Press.",
year = "2004",
month = jun,
day = "1",
doi = "10.1017/S0013091503000051",
language = "English",
volume = "47",
pages = "325--337",
journal = "Proceedings of the Edinburgh Mathematical Society",
issn = "0013-0915",
publisher = "Cambridge University Press",
number = "2",

}

RIS

TY - JOUR

T1 - On upper modular subalgebras of a Lie algebra.

AU - Bowman, Kevin

AU - Towers, David A.

AU - Varea, Vicente R.

N1 - RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics http://journals.cambridge.org/action/displayJournal?jid=UHY The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 47 (2), pp 325-337 2004, © 2004 Cambridge University Press.

PY - 2004/6/1

Y1 - 2004/6/1

N2 - This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper modular atom which is not an ideal. Finally it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a μ-algebra.

AB - This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper modular atom which is not an ideal. Finally it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a μ-algebra.

KW - Keywords

KW - Lie algebra

KW - subalgebra lattice

KW - upper modular

U2 - 10.1017/S0013091503000051

DO - 10.1017/S0013091503000051

M3 - Journal article

VL - 47

SP - 325

EP - 337

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -