TY - GEN

T1 - The geometry of extremal elements in a Lie algebra

AU - Cohen, A.M.

PY - 2012

Y1 - 2012

N2 - Let L be a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic distinct from 2 and from 3. Then L contains an extremal element, that is, an element x such that [x, [x, L]] is contained in the linear span of x in L. Suppose that L contains no sandwich, that is, no element x such that [x, [x, L]] = 0. Then, up to very few exceptions in characteristic 5, the Lie algebra L is generated by extremal elements and we can construct a building of irreducible and spherical type on the set of extremal elements of L. Therefore, by Tits’ classification of such buildings, L is determined by a known shadow space of a building. This gives a geometric alternative to the classical classification of finite-dimensional simple Lie algebras over the complex numbers and of classical finite-dimensional simple modular Lie algebras over algebraically closed fields of characteristic = 5. This paper surveys developments pertaining to this kind of approach to classical Lie algebras.

AB - Let L be a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic distinct from 2 and from 3. Then L contains an extremal element, that is, an element x such that [x, [x, L]] is contained in the linear span of x in L. Suppose that L contains no sandwich, that is, no element x such that [x, [x, L]] = 0. Then, up to very few exceptions in characteristic 5, the Lie algebra L is generated by extremal elements and we can construct a building of irreducible and spherical type on the set of extremal elements of L. Therefore, by Tits’ classification of such buildings, L is determined by a known shadow space of a building. This gives a geometric alternative to the classical classification of finite-dimensional simple Lie algebras over the complex numbers and of classical finite-dimensional simple modular Lie algebras over algebraically closed fields of characteristic = 5. This paper surveys developments pertaining to this kind of approach to classical Lie algebras.

U2 - 10.1007/978-1-4614-0709-6_2

DO - 10.1007/978-1-4614-0709-6_2

M3 - Conference contribution

SN - 978-1-4614-0708-9

T3 - Springer Proceedings in Mathematics (PROM)

SP - 15

EP - 35

BT - Buildings, finite geometries and groups

A2 - Narasimha Sastry, N.S.

PB - Springer

CY - New York

ER -