• Vijay Kodiyalam

Articles written in Proceedings – Mathematical Sciences

• The algebra ofG-relations

In this paper, we study a tower {AnG: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraAnG(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraAnG(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 {AnG(d): n} ≥ 1 (in the generic case).

• The planar algebra associated to a Kac algebra

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’.

• The planar algebra of a semisimple and cosemisimple Hopf algebra

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.

• From Graphs to Free Products

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.

• On a presentation of the spin planar algebra

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.

• Generators for finite depth subfactor planar algebras

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

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019