General linear Lie algebra

Let $𝕂$ be a field and $𝑛 \in ℕ$ . The general linear Lie algebra $𝔤 𝔩 𝑛 𝕂$ is the Lie algebra realized by $𝑛 \times 𝑛$ matrices under their linear commutator.¹ #m/def/lie More generally, if $𝑉$ is a vector space over $𝕂$ then $𝔤 𝔩 𝑉 = E n d 𝕂 𝑉$ is the commutator of the endomorphism ring $E n d 𝕂 𝑉$

[𝑒 𝑖 𝑗, 𝑒 𝑘 ℓ] = 𝛿 𝑗 𝑘 𝑒 𝑖 ℓ - 𝛿 𝑙 𝑖 𝑒 𝑘 𝑗

Properties

If $𝑥 \in E n d 𝑉 = 𝔤 𝔩 (𝑉)$ is nilpotent, then $a d 𝑥 \in D (𝔤 𝔩 (𝑉))$ is nilpotent.²

Proof

Consider the left- and right-regular representations of the $𝕂$ -monoid $E n d 𝑉$ , which we label $Λ$ and $P$ respectively. If $𝑥 \in E n d 𝑉$ is nilpotent, so too are $Λ (𝑥)$ and $P (𝑥)$ , whence $a d 𝑥 = Λ (𝑥) - P (𝑥)$ is nilpotent.

Triangular decomposition

$𝔤 𝔩 𝑛 𝕂$ has the archetypal triangular decomposition

𝔤 𝔩 𝑛 𝕂 = 𝔫 - \oplus 𝔥 \oplus 𝔫 +

where $𝔫 -$ consists of ^strictly-lower matrices, $𝔥$ consists of ^diagonal matrices, and $𝔫 +$ consists of ^strictly-upper matrices, i.e.

[𝔥, 𝔥] = 0 [𝔫 \pm, 𝔥] \subseteq 𝔫 \pm

Subalgebras

A subalgebra of $𝔤 𝔩 𝑛 𝕂$ is called a linear Lie algebra

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

1972. Introduction to Lie Algebras and Representation Theory, §1.2, p. 2 ↩
1972. Introduction to Lie Algebras and Representation Theory, §3.2, p. 12 ↩

General linear Lie algebra

Properties

Triangular decomposition

Subalgebras

Footnotes