# Andrew Sills

Affiliation: Georgia Southern University

Email: asills@georgiasouthern.edu

Title Of Talk: A classical q-hypergeometric approach to the $A_2^{(2)}$ standard modules

Abstract: 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 $A_1^{(1)}$. Later, they showed that in fact the Andrews-Gordon-Bressoud generalizations of the Rogers-Ramanujan identities explained" all the standard modules of all of $A_1^{(1)}$. The next logical step was to similarly try to explain $A_2^{(2)}$. 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 $A_2^{(2)}$ 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 $A_2^{(2)}$ standard modules as a whole, with a particular emphasis on those from levels 3 through 9.