$𝔰 𝔩 2 𝕂$

Let $𝕂$ be a field. $𝔰 𝔩 2 𝕂$ is the Lie algebra realized by traceless $2 \times 2$ matrices under their linear commutator. #m/def/lie It has the Chevalley basis

𝛼 1 = [10 0 - 1], 𝑥 𝛼 1 = [0100], 𝑥 - 𝛼 1 = [0010]

with the commutation relations

[𝛼 1, 𝑥 \pm 𝛼 1] = \pm 2 𝑥 \pm 𝛼 1 = ⟨ 𝛼 1, \pm 𝛼 1 ⟩ 𝑥 \pm 𝛼 1 [𝑥 𝛼 1, 𝑥 - 𝛼 1] = 𝛼 1

where we have the nondegenerate invariant symmetric bilinear form

⟨ 𝑥, 𝑦 ⟩ = t r 𝑥 𝑦 ⟨ 𝛼 1, 𝛼 1 ⟩ = 2 ⟨ 𝑥 𝛼 1, 𝑥 - 𝛼 1 ⟩ = 1 ⟨ 𝛼 1, 𝑥 \pm 𝛼 1 ⟩ = ⟨ 𝑥 \pm 𝛼 1, 𝑥 \pm 𝛼 1 ⟩ = 0

given by the trace form of the fundamental representation, making $𝔰 𝔩 2 𝕂$ a quadratic Lie algebra.

Gram matrix

Ordering the basis $(𝛼 1, 𝑥 𝛼 1, 𝑥 - 𝛼 1)$ , we have the following Gram matrix

⎡ ⎢ ⎢ ⎣ 200001010 ⎤ ⎥ ⎥ ⎦

Properties

Affine Lie algebras of $𝔰 𝔩 2 𝕂$

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

𝔰𝔩2⁡𝕂

Properties

$𝔰 𝔩 2 𝕂$