Hopf theory MOC

Restricted dual of a K-monoid

Let (A,πœ‡,πœ‚) be a K-monoid. Since the canonical inclusion

πœ„:Aβˆ—βŠ—Aβˆ—β†ͺ(AβŠ—A)βˆ—π‘“βŠ—π‘”β†¦π‘₯βŠ—π‘¦β†¦(𝑓π‘₯)(𝑔𝑦)

may not be epic, we cannot in general treat πœ‡βˆ— :Aβˆ— β†’(A βŠ—A)βˆ— as a comultiplication on Aβˆ—. To remedy this situation, we can construct the restricted dual as the preΓ―mage #m/def/hopf

Aπ‘œ=(πœ‡βˆ—)βˆ’1πœ„(Aβˆ—βŠ—Aβˆ—)

which is canonically a K-comonoid with comultiplication and coΓΌnit

πœ‡π‘œ:=πœ‡βˆ—β†ΎAπ‘œ,πœ‚π‘œ:=πœ‚βˆ—β†ΎAπ‘œ

respectively.

Proof of K-comonoid

#missing/proof

When A is a finite-dimensional 𝕂-monoid, πœ„ is an isomorphism and we call Aπ‘œ =Aβˆ— the dual.

Representative forms

Over a field 𝕂, the restricted dual is exactly the 𝕂-vector space of representative forms: The subspace of Aβˆ— spanned by the matrix coΓ«fficients of finite-dimensional A-modules. #m/thm/hopf

Proof

#missing/proof

#state/develop | #lang/en | #SemBr