Lie algebras MOC

Invariant bilinear form on a Lie algebra

A bilinear form ⟨ ⋅, ⋅⟩ :𝔤 ×𝔤 →𝕂 on a Lie algebra 𝔤 is said to be invariant iff the following equivalent conditions hold for any 𝑥,𝑦,𝑧 ∈𝔤¹

• ⟨[𝑥,𝑦],𝑧⟩ =⟨𝑥,[𝑦,𝑧]⟩

• ⟨ad𝑥⁡𝑦,𝑧⟩ +⟨𝑦,ad𝑥⁡𝑧⟩ =0

• ⟨[𝑥,𝑦],𝑧⟩ +⟨𝑦,[𝑥,𝑧]⟩ =0

See also

• Quadratic Lie algebra

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.6, p. 17 ↩