On Some Subfactors of Integer Index Arising from Vertex Models

Krishnan, Uma; Sunder, V. S.; Varughese, Cherian

doi:10.1006/jfan.1996.0114

Cited by 3 publications

(1 citation statement)

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“…By computing the first commuting square in Theorem A.10 we find that M −1 = C. This full Bernoulli shift over C with generator M 0 is called the ω-shift in [Rup95]. The corresponding subfactors have been investigated in [KSV96]. Note that these examples include the usual tensor shift (ω = 1) and the CAR-shift (ω = −1) in its Jordan-Wigner form [BR81].…”

Section: Some Concrete Examplesmentioning

confidence: 99%

Noncommutative Independence from the Braid Group $${\mathbb{B}_{\infty}}$$

Gohm

Köstler

2009

Commun. Math. Phys.

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We introduce 'braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group B∞. It provides an extension of exchangeability which is tied to the symmetric group S∞. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups Bn with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law.Furthermore we use the concept of product representations of endomorphisms [Goh04] with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of B∞ and the irreducible subfactor with infinite Jones index in the non-hyperfinite II 1 -factor L(B∞) related to it. Our investigations reveal a new presentation of the braid group B∞, the 'square root of free generator presentation' F 1/2 ∞ . These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.

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Section: Some Concrete Examplesmentioning

confidence: 99%