motivation of coalgebra

Question

I don't know why should we need coalgebra?What is the motivation?By changing all the arrows of algebra structure, it seems strange.What is the application of coalgebra? What is the relation with Lie algebra, representation theory, derived category? I am very confused about this definition.

Thank you.

Answer 1

Though there are many reasons why one could be interested in coalgebras, many researchers are interested in bialgebras (or Hopf algebras). Before you can understand these, a basic understanding of coalgebras is very advisable.

Let's consider some natural examples first.

Let $G$ be a finite group and consider the complex group algebra $\mathbb{C}[G]$. Consider the category $\mathcal{C}=\text{Rep}(G)$ of finite-dimensional $G$-representations ($\mathcal{C}=\mathbb{C}[G]-\text{mod}$). As you probably learned in a group representation theory course, one can take the tensor product of two $\mathbb{C}[G]$-modules $M$ and $N$. The tensor product $M\otimes_{\mathbb{C}}N$ is again a $\mathbb{C}[G]$-module and the action of $g\in G$ is given by $g\cdot (m\otimes n):=g\cdot m\otimes g\cdot n$.

Now consider a linear map $\Delta\colon\mathbb{C}[G]\to \mathbb{C}[G]\otimes_\mathbb{C} \mathbb{C}[G]$ determined by $\Delta(g)=g\otimes g$ for all $g\in G$. Note that this map encodes the action of $\mathbb{C}[G]$ on $M\otimes N$ in a natural way. Associativity of the tensor product naturally yields coassociativity for $\Delta$. Moreover, associativity of the $G$ action on $M\otimes_\mathbb{C} N$ forces the map $\Delta$ to be a algebra morphism, i.e. $\Delta(gh)=\Delta(g)\Delta(h)$. Hence the fact that $\mathcal{C}$ is a tensor category forces the algebra $\mathbb{C}[G]$ to be a bialgebra.

We can repeat te same story for Lie algebras.

Let $L$ be a nice complex finite-dimensional semisimple Lie algebra. Let $\mathcal{C}$ be the category of finite-dimensional Lie algebra representations of $L$. Equivalently, $\mathcal{C}=U(L)-\text{mod}$ where $U(L)$ is the universal enveloping algebra of $L$. Given two $L$-representations $M$ and $N$, the tensor product $M\otimes_\mathbb{C} N$ becomes a $L$-representation and the action is determined by $l\cdot (m\otimes n)=l\cdot m\otimes n+m\otimes l\cdot n$. Equivalenty, the action is encoded in the comultiplication $\Delta\colon U(L)\to U(L)\otimes_\mathbb{C} U(L)$ determined by $\Delta(l)=l\otimes 1+1\otimes l$ for all $l\in L\subset U(L)$. Once again, the tensor structure on $\mathcal{C}$ forces $U(L)$ to be a bialgebra.

After understanding these examples, you will quickly realize that a bialgebra $B$ yields a monoidal structure on its representation category. The converse is true as well to some extent (see the Tannaka reconstruction for more details).

As representation categories and monoidal structures have proven to be fundamental in many branches of physics and mathematics, I'd argue that these are strong reasons to be interested in coalgebras and bialgebras.