A class of objects—that are best described as being actions ofgroup-like objects of von Neumann algebras—is axiomatised and it is shown that there exists a bijective correspondence between isomorphism classes of suchcovariant systems and isomorphism classes of pairs of II₁ factors (M, N) satisfyingN⊂M, [M:N]<∞ andM∮N′=C.

In this paper, we study a tower {A_n^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA_n^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA_n^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A_n^G(d): n} ≥ 1 (in the generic case).

We define a certain abstract planar algebra by generators and relations, study the various aspects of its structure, and then identify it with Jones’ spin planar algebra.