Some Cogroup Objectss (Hopf Algebra) in CRing/Some Group Objects in Aff

Last time we talk about group object, cogroup object and representable functor(click here)

We did not give too many examples of cogroup object, that will be what I want to talk about in this essay.

We focus on the cogroup object in $\mathsf{CRing}$$\mathsf{CRing}$, so, they are Hopf Algebras.

We know that $\mathsf{Aff}\cong {\mathsf{CRing}}^{\mathsf{op}}$$\mathsf{Aff}\cong\mathsf{CRing^{op}}$, hence, take the Yoneda embedding $A⟼{\mathrm{Hom}}_{\mathsf{CRing}}(A,-)$$A\longmapsto\mathrm{Hom}_{\mathsf{CRing}}(A,-)$, we view the hot functor as affine scheme. If we could find some group object in $\mathrm{Aff}$$\mathrm{Aff}$, i.e. if ${\mathrm{Hom}}_{\mathsf{CRing}}(A,-)$$\mathrm{Hom}_{\mathsf{CRing}}(A,-)$ is a group object, then $A$$A$ is a cogroup object.

Here is some examples:

$$G{L}_n(-)\cong {\mathrm{Hom}}_{\mathsf{CRing}}(\mathbb{Z}[x_{1,1},...,x_{n,n}{]}_{\det },-)$$

$$S{L}_n(-)\cong {\mathrm{Hom}}_{\mathsf{CRing}}(\mathbb{Z}[x_{1,1},...,x_{n,n}]/(\det -1),-)$$

The unit group functor

$$U(-)\cong {\mathrm{Hom}}_{\mathsf{CRing}}(\mathbb{Z}[x,x^{-1}],-)$$

The Group of Roots of Unity

$${\mu }_n(-)\cong {\mathrm{Hom}}_{\mathsf{CRing}}(\mathbb{Z}[x]/(x^n-1),-)$$

Readers may remind this essay: Syntax and Semantics in Representable Functors

For $G{L}_n(-)$$GL_n(-)$ and $S{L}_n(-)$$SL_n(-)$, the correspondence co structure is

$$\mathrm{\Delta }(x_{i,j})=\sum _{r=1}^n{x}_{i,r}\otimes x_{r,j},\epsilon (x_{i,j})={\delta }_{i,j},S(X)=X^{-1}$$

$$S(x_{ij})=(X^{-1}{)}_{ij}=\frac{\mathrm{adj}(X{)}_{ij}}{\det \left(X\right)}=\frac{(-1{)}^{i+j}{M}_{ji}}{\det \left(X\right)}.$$

Let us check the diagram commute

$$\begin{array}{rl}\mathrm{\Delta }(x_{ij})& =\sum _{k=1}^n{x}_{ik}\otimes x_{kj},\\ (S\otimes \mathrm{id})(\mathrm{\Delta }(x_{ij}))& =\sum _{k=1}^{n}S(x_{ik})\otimes x_{kj}=\sum _{k=1}^n(X^{-1}{)}_{ik}\otimes x_{kj},\\ m((S\otimes \mathrm{id})\mathrm{\Delta }(x_{ij}))& =\sum _{k=1}^n(X^{-1}{)}_{ik}{x}_{kj}=(X^{-1}X{)}_{ij}={\delta }_{ij},\\ (\eta \circ \epsilon )(x_{ij})& =\eta (\epsilon (x_{ij}))=\eta ({\delta }_{ij})={\delta }_{ij}\cdot 1={\delta }_{ij}.\end{array}$$

For the unit group $U(-)$$U(-)$ and ${\mu }_n(-)$$\mu_n(-)$

$$\mathrm{\Delta }(x)=x\otimes x,\epsilon (x)=1,S(x)=x^{-1}$$

It is really easy to see that diagram commute.