A Classical Q-Hypergeometric Approach to the A2^(2) Standard Modules

In the early 1980's J. Lepowsky and R. Wilson gave the first Lie-theoretic proof of the Rogers-Ramanujan identities, and showed that they corresponded to the two inequivalent level 3 standard modules for the affine Kac-Moody Lie algebra A1^(1). Later, they showed that in fact the Andrews-Gordon-Bressoud generalizations of the Rogers-Ramanujan identities ``explained" all the standard modules of all of A1^(1). The next logical step was to similarly try to explain A2^(2). This has proved to be much more difficult. The level 2 modules correspond to a dilated version of the two Rogers-Ramanujan identities. In his 1988 Ph.D. thesis, Stefano Capparelli discovered a pair of new partition identities via his analysis of the level 3 standard modules. Further progress in this direction stalled until Debajyoti Nandi, in his 2014 PhD thesis, found the analogous set of three new (and very complicated) Rogers-Ramanujan type partition identities corresponding to the three inequivalent level 4 standard modules. The partition theoretic explanation of higher levels of A2^(2) continue to elude us. The history of partition identities are inexorably linked with that of q-series/q-product identities. In this talk, we examine a classical approach to q-hypergeometric identities that appear to be associated with the A2^(2) standard modules as a whole, with a particular emphasis on those from levels 3 through 9.