Algebra of Laurent polynomials

Let $𝕂$ be a field. The algebra $𝕂 [𝑡, 𝑡 - 1]$ of Laurent polynomials in indeterminate $𝑡$ is a $ℤ$ -graded commutative $𝕂$ -ring, with elements of the form

𝑓 = \sum 𝑛 \in ℤ 𝑓 𝑛 𝑡 𝑛

such that $𝑓 𝑛$ has finite support, with multiplication given by $𝑡 𝑛 \cdot 𝑡 𝑚 = 𝑡 𝑛 + 𝑚$ . It is isomorphic to the group algebra $𝕂 [ℤ]$ .

Properties

The degree derivation is given formally by $𝑑 = 𝑡 𝑑 𝑑 𝑡$
The derivations of $𝕂 [𝑡, 𝑡 - 1]$ form the Witt algebra.

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