### Abstract

An extremal element x in a Lie algebra g is an element for which the space [x,[x,g]] is contained in the linear span of x. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries on points and lines which can be characterized as root shadow spaces of buildings. In this paper we show that, in case the rank of this building is at least 3, the Lie algebra is (up to isomorphism) uniquely defined by this geometry. This provides us with a geometric characterization of (most of the) classical Lie algebras.

Original language | English |
---|---|

Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Journal of Algebra |

Volume | 502 |

DOIs | |

Publication status | Published - 15 May 2018 |

### Fingerprint

### Keywords

- Buildings
- Extremal elements
- Lie algebra

### Cite this

*Journal of Algebra*,

*502*, 1-23. https://doi.org/10.1016/j.jalgebra.2017.11.030

}

*Journal of Algebra*, vol. 502, pp. 1-23. https://doi.org/10.1016/j.jalgebra.2017.11.030

**A geometric characterization of the classical Lie algebras.** / Cuypers, Hans; Fleischmann, Yael.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A geometric characterization of the classical Lie algebras

AU - Cuypers, Hans

AU - Fleischmann, Yael

PY - 2018/5/15

Y1 - 2018/5/15

N2 - An extremal element x in a Lie algebra g is an element for which the space [x,[x,g]] is contained in the linear span of x. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries on points and lines which can be characterized as root shadow spaces of buildings. In this paper we show that, in case the rank of this building is at least 3, the Lie algebra is (up to isomorphism) uniquely defined by this geometry. This provides us with a geometric characterization of (most of the) classical Lie algebras.

AB - An extremal element x in a Lie algebra g is an element for which the space [x,[x,g]] is contained in the linear span of x. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries on points and lines which can be characterized as root shadow spaces of buildings. In this paper we show that, in case the rank of this building is at least 3, the Lie algebra is (up to isomorphism) uniquely defined by this geometry. This provides us with a geometric characterization of (most of the) classical Lie algebras.

KW - Buildings

KW - Extremal elements

KW - Lie algebra

UR - http://www.scopus.com/inward/record.url?scp=85044858901&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2017.11.030

DO - 10.1016/j.jalgebra.2017.11.030

M3 - Article

AN - SCOPUS:85044858901

VL - 502

SP - 1

EP - 23

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -