An inner product approach to Hopf algebras

Question

We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$.

Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^*: \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$ is a coproduct (coalgebraic operation)?

Is there a bialgebra structure whose product and coproduct are adjoints of each other? Is there a Hopf algebra with the latter property, and the additional condition that the antipode map is an isometry?

Note: The adjoint operators are taken with respect to the corresponding inner products.

Answer 1

This doesn't directly answer your question concerning Hopf structures on $\mathbb{C}^n$, but a particularly well-studied class of Hopf algebras for which the product is the adjoint of the coproduct are known as positive self-adjoint Hopf (PSH) algebras. These were originally introduced by Zelevinsky to study representations of finite classical groups. A concise, modern introduction can be found in notes from Grinberg and Reiner which have a focus on the application of PSH algebras to combinatorics.

To say a little more, a PSH algebra is a graded, connected Hopf algebra having a $\mathbb{Z}$ basis of homogeneous elements so that the structure constants for the algebra and coalgebra structures agree and are positive integers. Note that in any Hopf algebra with chosen basis the agreement of the structure constants for the algebra and coalgebra structures is equivalent to the condition that they are adjoint to one another.

The structure theorem of PSH algebras says that every PSH algebra is isomorphic to a tensor product of degree-shifts of the ring of symmetric functions, with one copy for each primitive basis element.