Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x\_1,\ldots, x\_n]\partial\_{x\_1}\oplus\ldots\oplus\mathbb{C}[x\_1,\ldots, x\_n]\partial\_{x\_n}$. My questions are as follows


Let $\mathcal{A}\otimes\_{\mathbb{C}}\mathbb{C}(t)$, $\mathcal{D}\otimes\_{\mathbb{C}}\mathbb{C}(t)$ are isomorphic $\mathbb{C}(t)$-Lie algebras.


1. Is it true that $\mathcal{A}, \mathcal{D}$ are isomorphic $\mathbb{C}$-Lie algebras?
2. Is it true that $\mathcal{A}\otimes\_{\mathbb{C}}\mathbb{C}[t], \mathcal{D}\otimes\_{\mathbb{C}}\mathbb{C}[t]$ are isomorphic $\mathbb{C}[t]$-Lie algebras?



Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?
Put $\mathcal{A}'=\mathcal{A}\otimes\_{\mathbb{C}}\mathbb{C}(t)$ and similarly for $\mathcal{D}'$. Choose an isomorphism $f\colon\mathcal{D}'\to\mathcal{A'}$.  

Choose a (countable) basis $\mathcal{D}\_0=\{d\_i:i\in\mathbb{N}\}$ for $\mathcal{D}$ over $\mathbb{C}$. Then $f(\mathcal{D}\_0)$ is a basis for $\mathcal{A}'$ over $\mathbb{C}(t)$ but $\dim\_{\mathbb{C}}(\mathcal{A}')=\dim\_{\mathbb{C}}(\mathcal{A})$ so we can also choose a countable basis $\mathcal{A}\_0=\{a\_i:i\in\mathbb{N}\}$ for $\mathcal{A}$ over $\mathcal{C}$. We must have $[d\_i,d\_j]=\sum\_kp\_{ijk}d\_k$ and $[a\_i,a\_j]=\sum\_kq\_{ijk}a\_k$ for some structure constants $p\_{ijk},q\_{ijk}\in\mathbb{C}$. Now let $K$ be a countable subfield of $\mathbb{C}$ containing all these structure constants, and also all the constants needed to ensure that $f(\mathcal{A}\_0)\subseteq K(t).\mathcal{D}\_0$ and $f^{-1}(\mathcal{D}\_0)\subseteq K(t).\mathcal{A}\_0$. Put $\mathcal{A}\_1=K.\mathcal{A}\_0$ and $\mathcal{D}\_1=K.\mathcal{D}\_0$, so these are Lie algebras over $K$ that become isomorphic over $K(t)$. If $\mathbb{C}$ were algebraic over $K$ then it would be countable, which is false. Thus, we can choose an embedding $i\colon K(t)\to\mathbb{C}$ extending the identity on $K$. By applying this to the coefficients of $f$, we obtain an isomorphism $\mathcal{D}\simeq\mathcal{A}$.


