nLab planar algebra

Idea

A planar algebra $\mathcal{P}$ consists of vector spaces $P_n$ 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 $R_i \in P_n$ to which linear combinations of planar tangles may be applied to obtain all elements of $\mathcal{P}$.

References

Details are given in

• Vaughan Jones, Quadratic Tangles in Planar Algebras (arXiv:1007.1158),

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.

• Vaughan Jones, Planar Algebra Master Class

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 $N$ 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

• Secret Blogging Seminar, Planar algebra entries