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.Comment: 16 pages, 20 figures; content adde

Guionnet-Jones-Shlyakhtenko subfactors associated to finite-dimensional Kac algebras

We analyse the Guionnet-Jones-Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.Comment: 18 pages, 21 figures, corrected typo

On Jones' planar algebras

We show that a certain natural class of tangles 'generate the collection of all tangles with respect to composition'. This result is motivated by, and describes the reasoning behind, the 'uniqueness assertion' in Jones' theorem on the equivalence between extremal subfactors of finite index and what we call 'subfactor planar algebras' here. This result is also used to identify the manner in which the planar algebras corresponding to M⊂M1 and Nop⊂Mop are obtained from that of N⊂M. Our results also show that 'duality' in the category of extremal subfactors of finite index extends naturally to the category of 'general' planar algebras (not necessarily finite-dimensional or spherical or connected or C∗, in the terminology of Jones)

The subgroup-subfactor

In this paper, we compute the standard invariant of the 'subgroup-subfactor' P×α|HH⊂P×αG, where α denotes an outer action of a finite group G on a II1 factor P, and P×α|HH denotes the obvi- ous crossed-product obtained by restricting the action to H. We then use this description to exhibit a pair of non-isomorphic subgroups Hi, i=1, 2, of the symmetric group S4 such that the subfactors R×α|Hi Hi⊂P×α G, i=1, 2 are conjugate, thereby disproving a conjecture of Thomsen-see [9]-that 'the subgroup-subfactor re- members the subgroup' (provided the subgroup contains no non-trivial normal subgroup of the ambient group)

From subfactor plannar algebras to subfactors

We present a purely planar algebraic proof of the main result of a paper of Guionnet-Jones-Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra

Spectra of principal graphs

We show that if the adjacency matrices of the two principal graphs of a finite index subfactor are regarded as (necessarily bounded, self-adjoint) operators on the ℓ2 spaces over their vertex sets, then their spectral measures, when restricted to the complement of {0}, are mutually absolutely continuous. In particular, for a finite-depth subfactor, the two matrices have the same sets of non-zero eigenvalues

Planar algebras and Kuperberg's 3-manifold invariant

We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a "planar algebra construction". A result of possibly independent interest, used during the proof, which relates duality in planar graphs and Hopf algebras, is the subject of a final section