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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?

3. Assume additionally that $\mathcal{A}\subseteq\mathcal{D}$. Is it true that $\mathcal{A}=\mathcal{D}$?

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?

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 $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}(t)\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}(t)$.

1. Is it true that $\mathcal{A}\cong\mathcal{D}$?

2. Is it true that $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}[t]\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}[t]$?

3. Assume additionally that $\mathcal{A}\subseteq\mathcal{D}$. Is it true that $\mathcal{A}=\mathcal{D}$?

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?

3. Assume additionally that $\mathcal{A}\subseteq\mathcal{D}$. Is it true that $\mathcal{A}=\mathcal{D}$?

Assume that $\mathcal{A}$ is $\mathbb{C}-$Lie algebra. Also denote the $\mathbb{C}-$Lie 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 $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}(t)\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}(t)$.

1. Is it true that $\mathcal{A}\cong\mathcal{D}$?

2. Is it true that $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}[t]\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}[t]$?

3. Assume additionally that $\mathcal{A}\subseteq\mathcal{D}$. Is it true that $\mathcal{A}=\mathcal{D}$?

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 $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}(t)\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}(t)$.

1. Is it true that $\mathcal{A}\cong\mathcal{D}$?

2. Is it true that $\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}[t]\cong\mathcal{D}\otimes_{\mathbb{C}}\mathbb{C}[t]$?

3. Assume additionally that $\mathcal{A}\subseteq\mathcal{D}$. Is it true that $\mathcal{A}=\mathcal{D}$?