General question:
Let $V$ be a finite dimensional vector space. It is clear that $V\cong \mathbb{F}^{\dim V}=\mathbb{A}_{\mathbb F}^{\dim V}$, and so you might want to give $V$ a Zariski topology based on the linear isomorphism (or equivalently, the basis of $V$). Is there a genral way to do that?
The problem, as I see it, is that you can have many different isomorphisms and therefore different topologies.
Specific Question:
I'm reading about lie groups $G$, their associated lie algebras $\mathfrak{g}$, and the exterior algebra $\Lambda^d(\mathfrak{g})$ for some $d\leq \dim \mathfrak{g}$. In a paper I'm reading, there is a use (without explanation) of the terms of Zariski topology regarding $\Lambda^d(\mathfrak{g})$. What is the correct interpretation of this?