planar algebra



A planar algebra 𝒫\mathcal{P} consists of vector spaces P nP_n together with multilinear operations between them indexed by planar tangles – large discs with internal (“input”) discs all connected up by non-intersecting curves called strings. Thus a planar algebra may be thought of as made up from generators R iP nR_i \in P_n to which linear combinations of planar tangles may be applied to obtain all elements of 𝒫\mathcal{P}.


Details are given in

where the author remarks

The definition of a planar algebra has evolved a bit since the original one in Planar algebras I so we give a detailed definition which is, we hope, the ultimate one, at least for shaded planar algebras.

Relations will appear with knot theory, statistical mechanics, combinatorial group theory, quantum groups and most of all, subfactors. They lead to many different algebraic structures according to the particular planar tangles chosen. Of particular interest recently have been graded algebra structures which give connections with the large NN limit of random matrices and free probability.

The relation to subfactors and fusion categories is discussed in

  • Scott Morrison, Small index subfactors, planar algebras, and fusion categories, lectures at CT2013 (pdf)

Plenty of discussion of planar algebras is at

Last revised on July 3, 2013 at 18:55:01. See the history of this page for a list of all contributions to it.