Many noncommutative algebraists in the 1980s were aware of the successful marriage of algebra and algebraic geometry in the commutative setting and wished to duplicate that relationship in the noncommutative setting. One such line of study was the search for a subclass of noncommutative algebras that “behave” enough like polynomial rings that a geometric theory could be developed for them. One proposal for such a class of algebras are the regular algebras, introduced in [Artin and Schelter 1987], that were investigated using new geometric techniques in the pivotal papers of M. Artin, J. Tate and M. Van den Bergh [Artin et al. 1990; 1991].

About the same time, advances in quantum mechanics in the 20th century had produced many new noncommutative algebras on which traditional techniques had only yielded limited success, so a need had arisen to find new techniques to study such algebras (see [Reshetikhin et al. 1989; Kapustin et al. 2001; Sklyanin 1982; 1985; Sudbery 1993]). One such algebra was the Sklyanin algebra, which had emerged from the study of quantum statistical mechanics [Sklyanin 1982; 1985]. By the early 1990s, T. Levasseur, S. P. Smith, J. T. Stafford and others had solved the ten-year old open problem of completely classifying all the finite-dimensional irreducible representations (simple modules) over the Sklyanin algebra, and their methods were the geometric techniques developed by Artin, Tate and Van den Bergh [Levasseur and Smith 1993; Smith and Stafford 1992; Smith and Staniszkis 1993].