Algebraic properties of quantum quasigroups

Linear aspects of equational triality in quasigroups

“…Therefore, since triality is an equational theory, in order for our definition to be an honest generalization, it has to exist in noncoassociative and noncocommutative settings. This chapter extends the work of [25] in two ways: 1.) proving that H-bialgebras exhibit conjugate triality (Prop.…”

“…In this paper, we defined a notion of conjugates from quantum quasigroups. As an extension of the results from our paper [25], we provide two examples -Pérez-Izquierdo's H-bialgebras and Smith's quantum couple-of quantum quasigroup constructions which exhibit triality in their set of conjugates. (b) While quasigroup divisions allow us to define a universal algebraic variety of quasigroups, and are a necessary component of the triality theory, knowledge of the multiplication specifies the entire structure.…”

“…Chapter 5 examines a generalization of the triality theory to symmetric monoidal categories. Some of this is joint work with Im and Smith [25]. The transportation of groups into categories of vector spaces via Hopf algebras has produced an explosion of results, unifying ostensibly unrelated areas such as statistical mechanics and knot theory (cf.…”

“…Chapter 5 expands on some of the work in [25]. In this paper, we defined a notion of conjugates from quantum quasigroups.…”

“…We described a theory of conjugates for quantum quasigroups introduced by the author, Im, and Smith in [25]. To justify the theory's suitability for symmetric monoidal categories, we proved that a noncocommutative, noncoassociative quantum quasigroup structure on the smash product Aut(Q)#K(Q) led to a collection of conjugates on which S 3 acts.…”