Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and $\mathbb{C}[X]$ be the algebra of functions on $X$. It is said that the action of $U$ on $X$ induced a co-action of $\mathbb{C}[U]$ on $\mathbb{C}[X]$ and an action of $U(\mathfrak{n})$ on $\mathbb{C}[X]$. Where $\mathfrak{n}$ is the Lie algebra of $U$ and $U(\mathfrak{n})$ is the universal enveloping algebra of $\mathfrak{n}$. Therefore we have two maps $\varphi: \mathbb{C}[X] \to \mathbb{C}[U] \otimes \mathbb{C}[X]$ and $\psi: U(\mathfrak{n}) \otimes \mathbb{C}[X] \to \mathbb{C}[X] $.
My question is: how to define $\varphi$ and $\psi$ using the action $f: U \times X \to X$? Thank you very much.
Edit: now I know the action $\psi$ is given by the following formula: for $E \in U(\mathfrak{n})$, $f \in \mathbb{C}[X]$, we have $$ E(f)(x) = \frac{d}{dt} |_{t=0} f(e^{tE}\cdot x), $$ here $e^{tE}\cdot x$ means $e^{tE} \in U$ acts on $x$.
But I don't know the coation $\varphi: \mathbb{C}[X] \to \mathbb{C}[U] \otimes \mathbb{C}[X]$.
Let $\lambda: U \times X \to X$ denote the left action.
Given $f \in \Bbb{C}[X]$ and $u \in U$, $x \in X$, define $\varphi(f) \in \mathbb{C}[U] \otimes \mathbb{C}[X]$ by $$ \varphi(f)(u \otimes x) = f(\lambda(u, x)) = f(ux). $$
Here we have implicitly identified $\mathbb{C}[U] \otimes \mathbb{C}[X] \overset{\cong}{\to} \mathbb{C}[U \times X]$.