Rogers-Ramanujan exact sequences and free representations over free generalized vertex algebras

The Rogers-Ramanujan recursions are studied from the viewpoint of free representations over free (generalized) vertex algebras. Specifically, we construct short exact sequences among the free representations over free generalized vertex algebras which lift the recursions. They naturally generalize exact sequences introduced by S. Capparelli et al. We also show that an analogue of the Rogers-Ramanujan recursions is realized as an exact sequence among finite-dimensional free representations over a certain finite-dimensional vertex algebra. As an application of exact sequences, we reveal a relation between free generalized vertex algebras and $\hat{\mathfrak{sl}}_2$ spaces of coinvariants $L_{1,0}^{N,\infty}(\mathfrak n)$ introduced by B. Feigin et al. Moreover, it is shown that generalized $q$-Fibonacci recursions may be realized as exact sequences obtained by applying several functors to exact sequences among free representations over free generalized vertex algebras. Finally, we consider a character decomposition formula of the basic $\hat{\mathrm{sl}}_2$-module $L_{1,0}$ by M. Bershtein et al. related to the Urod vertex operator algebras, from the viewpoint of free representations and exact sequences.