In Toen's *Affine Stacks*, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we choose dg algebras as a model for derived rings). To prove this, he explicitly constructs a cosimplicial ring whose $\text{Spec}$ is $B\mathbb{G}_a$, verifying this by showing they satisfy the same universal property.

I'm wondering if there's a more explicit way to do this calculation. In particular, I would like to compute $\mathcal{O}(BG)$ for various linear algebraic $G$ (unipotent, solvable, semisimple...). I would also like to compute $\mathcal{O}(G/G)$ (the adjoint action).

Here's an idea for an approach, which might be totally wrong. These stacks are Artin stacks, so in particular one can write $BG$ as a colimit of a simplicial diagram in schemes. Then by definition, $\mathcal{O}(BG)$ is a limit of cosimplicial diagram in algebras. Here I'm not exactly sure what I'm doing, but I want to say I can apply some kind of (dual) monoidal Dold-Kan to realize this limit in dg algebras. I think (not sure) one only gets a commutative algebra structure up to homotopy here, but at least taking cohomology, this ends up just being the Hochschild complex for computing the ("rational") cohomology of the (rational) trivial representation of $G$.

Using this, I think one computes that $\mathcal{O}(B\mathbb{G}_m) = k$ (but haven't verified the details), and Jantzen's book (Representations of Algebraic Groups) verifies Toen's result for $\mathbb{G}_a$. Very quickly though, this computation becomes difficult. Are there general results/computations known for unipotent, solvable, or reductive $G$, and can one compute $\mathcal{O}(G/G)$ in a similar way (e.g. using the rational cohomology of $k[G]$ under the adjoint action)? Do you have a preferred way to think about this?