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 #m/def/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.2
Proof

Consider the left- and right-regular representations of the -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

#state/develop | #lang/en | #SemBr

Footnotes

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

  2. 1972. Introduction to Lie Algebras and Representation Theory, §3.2, p. 12 ↩