Special linear Lie algebra

𝔰𝔩2⁡𝕂

Let 𝕂 be a field. 𝔰𝔩2⁡𝕂 is the Lie algebra realized by traceless 2 ×2 matrices under their linear commutator. lie It has the Chevalley basis

𝛼1=[100−1],𝑥𝛼1=[0100],𝑥−𝛼1=[0010]

with the commutation relations

[𝛼1,𝑥±𝛼1]=±2𝑥±𝛼1=⟨𝛼1,±𝛼1⟩𝑥±𝛼1[𝑥𝛼1,𝑥−𝛼1]=𝛼1

where we have the ^nondegenerate invariant symmetric bilinear form

⟨𝑥,𝑦⟩=tr⁡𝑥𝑦⟨𝛼1,𝛼1⟩=2⟨𝑥𝛼1,𝑥−𝛼1⟩=1⟨𝛼1,𝑥±𝛼1⟩=⟨𝑥±𝛼1,𝑥±𝛼1⟩=0

given by the Trace form of the fundamental representation, making 𝔰𝔩2⁡𝕂 a quadratic Lie algebra.

Gram matrix

Ordering the basis (𝛼1,𝑥𝛼1,𝑥−𝛼1), we have the following Gram matrix

⎡⎢ ⎢⎣200001010⎤⎥ ⎥⎦

Properties

• Affine Lie algebras of sl_2

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