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Laurent pole projection satisfies the weight-minus-one identity

Statement

Let $K$ be a commutative unital $Q$ -algebra, let $ϵ$ be formal, and put $A = K ((ϵ))$ with projection $R$ onto strictly negative powers. For all $x, y \in A$ , $R$ satisfies $R (x) R (y) = R (R (x) y + x R (y) - x y) .$

Statement references

Laurent-series target and pole projection for subtraction

Facts & Assumptions

Given: The Laurent-series algebra and projection of the Definition.

[F1]

Laurent-series target and pole projection for subtraction: $A = A_{-} \oplus A_{+}$ , $R$ projects onto $A_{-}$ , $A_{-}$ is closed under multiplication, and $A_{+}$ is closed under multiplication.

Proof

Given: $x, y \in A$ and [F1].

1.1givenF1algebra

Write $x = x_{-} + x_{+}$ and $y = y_{-} + y_{+}$ using the direct sum in [F1], so $R (x) = x_{-}$ and $R (y) = y_{-}$ . Then $R (x) y + x R (y) - x y = x_{-} (y_{-} + y_{+}) + (x_{-} + x_{+}) y_{-} - (x_{-} + x_{+}) (y_{-} + y_{+}) = x_{-} y_{-} - x_{+} y_{+} .$

2.1F1step 1.1algebra∎

Since $x_{-} y_{-} \in A_{-}$ and $x_{+} y_{+} \in A_{+}$ , applying $R$ to the identity in step 1.1 gives $R (R (x) y + x R (y) - x y) = x_{-} y_{-} = R (x) R (y)$ .

Depends on

Laurent-series target and pole projection for subtraction

Dependency tree · 1 level

Sources

Dominique Manchon, Hopf algebras, from basics to applications to renormalization, full lecture notes, arXiv:math/0408405v2