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 obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.

We investigate a construction (from Kodiyalam Vijay and Sunder V S, J. Funct. Anal.260 (2011) 2635–2673) which associates a finite von Neumann algebra $M(\Gamma, \mu)$ to a finite weighted graph $(\Gamma, \mu)$. Pleasantly, but not surprisingly, the von Neumann algebra associated to a `flower with 𝑛 petals’ is the group on Neumann algebra of the free group on 𝑛 generators. In general, the algebra $M(\Gamma, \mu)$ is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs `with one edge’ (or actually a pair of dual edges). This also yields `natural’ examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) $\mathbb{C}\oplus\mathbb{C}$-valued circular and semi-circular operators.

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.

We show that a subfactor planar algebra of finite depth $k$ is generated by a single $s$-box, for $s\leq {\rm min}\{ k + 4, 2k\}$.