Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces.

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 (so 5) , L 1 (so 8) and L 1 (g 2). We derive a new character formula for the principal subspace of L 1 (so 5) , proving that it is classically free, and present evidence that the principal subspaces of L 1 (so 8) and of L 1 (g 2) are also classically free. [ABSTRACT FROM AUTHOR]