Antipode

The antipode $S$ , the last piece of a Hopf algebra: the unique map with $m(S \otimes \mathrm{id})\Delta = \eta\varepsilon$ .

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Antipode(element)

S(x)

Typed signatures

Type	Package	Applies
`(number) -> number`	`hopf`	always

Domain: Hopf algebras

Details

On a graded connected Hopf algebra the axiom DETERMINES $S$ : splitting off the two trivial terms of $\Delta$ gives $S(x) = -x - \sum S(x')x''$ , and the left factor's degree strictly drops, so the recursion terminates
Computed that way and then verified by running the axiom, rather than trusted
On a commutative Hopf algebra $S$ is an involution, so $S^2 = \mathrm{id}$ on QSym

Plot

Basic #

$S(M_1) = -M_1$

\mathrm{Antipode}(\mathrm{QSymM}(\bigl\lbrack1\bigr\rbrack))

Out

-\mathrm{QSymM}(\bigl\lbrack1\bigr\rbrack)

Properties #

$S^2 = \mathrm{id}$ , since QSym is commutative

\mathrm{Antipode}(\mathrm{Antipode}(\mathrm{QSymM}(\bigl\lbrack1, 2\bigr\rbrack)))

Out

\mathrm{QSymM}(\bigl\lbrack1, 2\bigr\rbrack)

the product is homogeneous, so degrees add

\mathrm{HopfDegree}(\mathrm{NonCommutativeMultiply}(\mathrm{QSymM}(\bigl\lbrack1\bigr\rbrack), \mathrm{QSymM}(\bigl\lbrack2\bigr\rbrack)))

Out

3

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