Loop algebra

Let $𝔤$ be a Lie algebra over $𝕂$ . The loop algebra $𝔤 [𝑡, 𝑡 - 1]$ of $𝔤$ is the tensor product algebra $𝔤 \otimes 𝕂 [𝑡, 𝑡 - 1]$ where $𝕂 [𝑡, 𝑡 - 1]$ is the algebra of Laurent polynomials, #m/def/lie i.e. with the bracket

[𝑥 \otimes 𝑓, 𝑦 \otimes 𝑔] = [𝑥, 𝑦] \otimes 𝑓 𝑔

for any $𝑥, 𝑦 \in 𝔤$ and $𝑓, 𝑔 \in 𝕂 [𝑡, 𝑡 - 1]$ . This may also be viewed as formal series $𝔤 [𝑡, 𝑡 - 1]$ .

Loop algebra

See also