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On Lie algebras all of whose minimal subalgebras are lower modular.

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On Lie algebras all of whose minimal subalgebras are lower modular. / Bowman, Kevin; Towers, David A.; Varea, Vicente R.
In: Communications in Algebra, Vol. 32, No. 12, 2004, p. 4515-4533.

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

Harvard

Bowman, K, Towers, DA & Varea, VR 2004, 'On Lie algebras all of whose minimal subalgebras are lower modular.', Communications in Algebra, vol. 32, no. 12, pp. 4515-4533. https://doi.org/10.1081/AGB-200036762

APA

Bowman, K., Towers, D. A., & Varea, V. R. (2004). On Lie algebras all of whose minimal subalgebras are lower modular. Communications in Algebra, 32(12), 4515-4533. https://doi.org/10.1081/AGB-200036762

Vancouver

Bowman K, Towers DA, Varea VR. On Lie algebras all of whose minimal subalgebras are lower modular. Communications in Algebra. 2004;32(12):4515-4533. doi: 10.1081/AGB-200036762

Author

Bowman, Kevin ; Towers, David A. ; Varea, Vicente R. / On Lie algebras all of whose minimal subalgebras are lower modular. In: Communications in Algebra. 2004 ; Vol. 32, No. 12. pp. 4515-4533.

Bibtex

@article{d430d6f567034fd7b46ef77c8f5044a2,
title = "On Lie algebras all of whose minimal subalgebras are lower modular.",
abstract = "The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.",
keywords = "Lie algebras, Lattice of subalgebras, Modular subalgebra",
author = "Kevin Bowman and Towers, {David A.} and Varea, {Vicente R.}",
note = "The final, definitive version of this article has been published in the Journal, Communications in Algebra, 32 (12), 2004, {\textcopyright} Informa Plc",
year = "2004",
doi = "10.1081/AGB-200036762",
language = "English",
volume = "32",
pages = "4515--4533",
journal = "Communications in Algebra",
issn = "0092-7872",
publisher = "Taylor and Francis Ltd.",
number = "12",

}

RIS

TY - JOUR

T1 - On Lie algebras all of whose minimal subalgebras are lower modular.

AU - Bowman, Kevin

AU - Towers, David A.

AU - Varea, Vicente R.

N1 - The final, definitive version of this article has been published in the Journal, Communications in Algebra, 32 (12), 2004, © Informa Plc

PY - 2004

Y1 - 2004

N2 - The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.

AB - The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.

KW - Lie algebras

KW - Lattice of subalgebras

KW - Modular subalgebra

U2 - 10.1081/AGB-200036762

DO - 10.1081/AGB-200036762

M3 - Journal article

VL - 32

SP - 4515

EP - 4533

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 12

ER -