  We analyze the structure of Feigin-Stoyanovsky's principal subspaces of
affine Lie algebra from the jet algebra viewpoint. For type $A$ level one
principal subspaces, we show that their shifted multi-graded Hilbert series can
be expressed either using the quantum dilogarithm or as certain generating
functions ``counting" finite-dimensional representations of $A$-type quivers.
This notably results in novel fermionic character formulas for these principal
subspaces. Moreover, our result implies that all level one principal subspaces
of type $A$ are ``classically free" as vertex algebras.
  We also analyze infinite jet algebras associated to principal subspaces of
affine vertex algebras $L_{1}(\mathfrak{so}_5)$, $L_{1}(\mathfrak{so}_8)$ and
$L_1(\frak{g}_2)$. We derive a new character formula for the principal subspace
of $L_1(\mathfrak{so}_5)$, proving that it is classically free, and present
evidence that the principal subspaces of $L_1(\mathfrak{so}_8)$ and of
$L_1(\frak{g}_2)$ are also classically free.
