Lie algebras MOC

General linear Lie algebra

Let be a field and . The general linear Lie algebra is the Lie algebra realized by matrices under their linear commutator.¹ lie More generally, if is a vector space over then is the commutator of the endomorphism ring

Properties

1. If is nilpotent, then is nilpotent.^[1972. Introduction to Lie Algebras and Representation Theory, §3.2, p. 12]

Proof

Consider the left- and right-regular representations of the K-monoid , which we label and respectively. If is nilpotent, so too are and , whence is nilpotent.

Triangular decomposition

has the archetypal triangular decomposition

where consists of ^strictly-lower matrices, consists of ^diagonal matrices, and consists of ^strictly-upper matrices, i.e.

Subalgebras

A subalgebra of is called a linear Lie algebra

• Special linear Lie algebra
• Lie algebra of nilpotent endomorphisms

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §1.2, p. 2 ↩