V S Sunder
Articles written in Proceedings – Mathematical Sciences
Volume 113 Issue 1 February 2003 pp 15-51
The planar algebra associated to a Kac algebra
Vijay Kodiyalam Zeph Landau V S Sunder
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’.
Volume 116 Issue 4 November 2006 pp 373-373
Volume 116 Issue 4 November 2006 pp 443-458 Operator Theory/Operator Algebras/Quantum Invariants
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.
Volume 122 Issue 4 November 2012 pp 547-560
Madhushree Basu Vijay Kodiyalam V S Sunder
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.
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