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## "Index for subfactors", Vaughan Jones, 1983

*Reviewed September 1, 2023*

*Citation:* Jones, Vaughan FR. "Index for subfactors." Inventiones mathematicae 72.1 (1983): 1-25.

*Web:* https://link.springer.com/article/10.1007/BF01389127

*Tags:* Foundational, Mathematical, Hadamard-matrices, Subfactors

This is the paper which introduces the notion of a subfactor,
and shows that it takes its characteristic discrete-then-continuous set of values.

This is a foundational paper for a lot of reasons, and it sparked a lot of interest in subfactors. It is this paper
which noticed the connection between Temperley-Lieb-Jones algebras and subfactors. TLJ algebras were introduced in

> Temperley, Harold NV, and Elliott H. Lieb. "Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolationâ€™problem." Condensed Matter Physics and Exactly Soluble Models: Selecta of Elliott H. Lieb (2004): 475-504.

A nice discussion of the relevance of this paper is found on MathOverflow,
here.

There's a nice immediate connection between factors and modular tensor categories.
The non-degeneracy condition on an MTC says that there are no transparent particles - everybody commutes non-trivially with somebody.
The factor condition on a von Neumann algebra says that every operator in the algebra must commute non-trivially with somebody. Really, these are the same condition!