Modular Lie Algebras pp 73-95 | Cite as

# Forms of the Classical Lie Algebras

## Abstract

The discussion in this chapter may be regarded as motivated by the problem of determining all Lie algebras L with non-singular Killing form over an arbitrary field F (characteristic ≠ 2, 3). By Chapter I, § 7, the problem reduces to the case where 1D50F is simple. If ℨ is the centroid of L, then L is normal simple when regarded as a Lie algebra over ℨ moreover, if *S* is a ℨ-linear transformation of L, we have Tr_{F} (*S*) = *T*_{ℨ/F} (Tr_{ℨ} (*S*)), where *T*_{ℨ/F} is the trace in the field extension ℨ/F (cf. [223, p. 66]). It follows that L has non-singular Killing form over ℨ, and that _{T
ℨ/F} is not zero, hence that ℨ/F is a separable extension. Thus the problem is reduced to the study of finite separable extensions of F, and to the case where L is normal simple. Assuming L normal simple over F, let *Ω* be an algebraically closed extension of F. Then L_{
Ω } has non-singular Killing form, and so is a classical simple Lie algebra over *Ω* therefore L
_{ Ω }
belongs to one of an already determined set of isomorphism classes. The “problem of forms” is that of describing L when L
_{ Ω }
is known, i.e., of determining the F-isomorphism classes of algebras L such that L
_{ Ω }
belongs to a given *Ω-*isomorphism class.

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