I have a question which is not really precise, unfortunately.
Let $A$ be a Poisson $n$-algebra, i.e. a graded commutative algebra with a Lie bracket of degree $n-1$ s.t. the bracket is a biderivation w.r.t. the commutative product. Add a new square-zero variable $\varepsilon$ to $A$, possibly of positive degree, to get $A[\varepsilon]$. Clearly this is a graded commutative algebra, and there is also a shifted Lie bracket $[a+\varepsilon b, a'+\varepsilon b'] := [a,a'] + \varepsilon ([a,b'] \pm [a',b])$.
If $A$ has some geometric meaning (and this is the imprecise part), does $A[\varepsilon]$ also has some geometric meaning? I'm open to various notions of "geometric meaning". I found this appearing in an algebro-topological setting, and I'm wondering if there is a bigger picture I don't know about. This may be something very simple. My best guess is that $A[\varepsilon]$ is twice the dimension of $A$, so perhaps this is related to some (co)tangent bundle...?