Lie algebras MOC

Loop algebra

Let 𝔤 be a Lie algebra over 𝕂. The loop algebra 𝔤[𝑡,𝑡−1] of 𝔤 is the tensor product algebra 𝔤 ⊗𝕂[𝑡,𝑡−1] where 𝕂[𝑡,𝑡−1] is the algebra of Laurent polynomials, lie i.e. with the bracket

[𝑥⊗𝑓,𝑦⊗𝑔]=[𝑥,𝑦]⊗𝑓𝑔

for any 𝑥,𝑦 ∈𝔤 and 𝑓,𝑔 ∈𝕂[𝑡,𝑡−1]. This may also be viewed as formal series 𝔤[𝑡,𝑡−1].

See also

• Affine Lie algebra
• Twisted affine Lie algebra

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