Communications in Mathematical Physics

, Volume 339, Issue 2, pp 729–768 | Cite as

Infinite Index Subfactors and the GICAR Categories

  • Vaughan F. R. Jones
  • David Penneys


Given a II1-subfactor \({A \subset B}\) of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as AA bimodule maps among the bimodules \({\bigotimes_A^n L^2(B)}\). As a corollary, we get a lower bound on the dimension of the centralizer algebras \({A_{0}^{'}\,\cap\,A_{2n}}\) for infinite index subfactors, and we also get that \({A_{0}^{'}\,\cap\,A_{2n}}\) is nonabelian for \({n \geq 2}\), where \({(A_n)_{n \geq 0}}\) is the Jones tower for \({A_0 = A \subset B = A_1}\). We also show that the annular GICAR/planar rook category acts as maps amongst the A-central vectors in \({\bigotimes_A^n L^2(B)}\), although this action may be degenerate. We prove these results in more generality using bimodules. The embedding of the GICAR category builds on work of Connes and Evans, who originally found GICAR algebras inside Temperley–Lieb algebras with finite modulus.


Tensor Category Minimal Projection Bratteli Diagram Centralizer Algebra Cellular Algebra 
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  1. BDH11.
    Bartels, A., Douglas, C.L., Henriques, A.: Dualizability and index of subfactors. Quantum Topol. 5(3) (2014). arXiv:1110.5671
  2. Big12.
    Bigelow, S.: A diagrammatic Alexander invariant of tangles. J. Knot Theory Ramif. 21(8), 1250081, 9 (2012). doi: 10.1142/S0218216512500812. arXiv:1203.5457
  3. BRY12.
    Bigelow, S., Ramos, E., Yi, R.: The Alexander and Jones polynomials through representations of rook algebras. J. Knot Theory Ramif. 21(12), 1250114, 18 (2012). doi: 10.1142/S0218216512501143. arXiv:1110.0538
  4. Bur03.
    Burns, M.: Subfactors, planar algebras, and rotations, Ph.D. thesis, University of California, Berkeley (2003). arXiv:1111.1362
  5. CE89.
    Connes, A., Evans, D.E.: Embedding of U(1)-current algebras in noncommutative algebras of classical statistical mechanics. Commun. Math. Phys. 121(3), 507–525 (1989)Google Scholar
  6. Con80.
    Connes, A.: On the spatial theory of von Neumann algebras. J. Funct. Anal. 35(2), 153–164 (1980)Google Scholar
  7. Dav96.
    Davidson K.R.: C *-algebras by example. In: Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)Google Scholar
  8. EN96.
    Enock, M., Nest, R.: Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras. J. Funct. Anal. 137(2), 466–543 (1996)Google Scholar
  9. FHH09.
    Flath, D., Halverson, T., Herbig, K.: The planar rook algebra and Pascal’s triangle. Enseign. Math. (2) 55(1–2), 77–92 (2009). arXiv:0806.3960
  10. GL96.
    Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)Google Scholar
  11. GL98.
    Graham, J.J., Lehrer, G.I.: The representation theory of affine Temperley–Lieb algebras. Enseign. Math. (2) 44(3–4), 173–218 (1998), 173–218 (1998)Google Scholar
  12. Haa79.
    Haagerup, U.: Operator-valued weights in von Neumann algebras. I. J. Funct. Anal. 32(2), 175–206 (1979)Google Scholar
  13. HO89.
    Herman R.H., Ocneanu A.: Index theory and Galois theory for infinite index inclusions of factors. C. R. Acad. Sci. Paris Sér. I Math. 309(17), 923–927 (1989)MathSciNetGoogle Scholar
  14. ILP98.
    Izumi M., Longo R., Popa S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155(1), 25–63 (1998)MathSciNetCrossRefGoogle Scholar
  15. Jon83.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983). doi: 10.1007/BF01389127 MathSciNetADSCrossRefGoogle Scholar
  16. Jon99.
    Jones V.F.R.: Planar algebras I (1999). arXiv:math/9909027
  17. Jon01.
    Jones, V.F.R.: The annular structure of subfactors. In: Essays on Geometry and Related Topics, vol. 1, 2; Monogr. Enseign. Math., vol. 38, pp. 401–463. Enseignement Math., Geneva (2001)Google Scholar
  18. Jon10.
    Jones, V.F.R.: Von Neumann algebras. (2010). Accessed 13 June 2015
  19. Jon11.
    Jones, V.F.R.: Jones’ notes on planar algebras. (2011). Accessed 13 June 2015
  20. Kau87.
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987). doi: 10.1016/0040-9383(87)90009-7
  21. Lie67.
    Lieb, E.H.: Residual entropy of square ice. Phys. Rev. 162(1), 162–172 (1967)Google Scholar
  22. Ocn88.
    Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator Algebras and Applications, vol. 2. London Math. Soc. Lecture Note Ser., vol. 136, pp. 119–172. Cambridge Univ. Press, Cambridge (1988)Google Scholar
  23. Ocn94.
    Ocneanu, A.: Chirality for operator algebras. In: Subfactors (Kyuzeso, 1993), pp. 39–63. World Sci. Publ., River Edge (1994)Google Scholar
  24. Pen12.
    Penneys D.: A cyclic approach to the annular Temperley–Lieb category. J. Knot Theory Ramif. 21(6), 1250049, 40 (2012). doi: 10.1142/S0218216511010012. arXiv:0912.1320
  25. Pen13.
    Penneys D.: A planar calculus for infinite index subfactors. Commun. Math. Phys. 319(3), 595–648 (2013). doi: 10.1007/s00220-012-1627-4. arXiv:1110.3504
  26. Pop86.
    Popa, S.: Correspondences, INCREST Preprint (1986)Google Scholar
  27. Pop93.
    Popa, S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111(2), 375–405 (1993). doi: 10.1007/BF01231293
  28. Pop95.
    Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120(3), 427–445 (1995). doi: 10.1007/BF01241137 MathSciNetADSCrossRefGoogle Scholar
  29. Sol02.
    Solomon L.: Representations of the rook monoid. J. Algebra 256(2), 309–342 (2002). doi: 10.1016/S0021-8693(02)00004-2 MathSciNetCrossRefGoogle Scholar
  30. TL71.
    Temperley H.N.V., Lieb E.H.: Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem. Proc. R. Soc. Lond. Ser. A 322(1549), 251–280 (1971)MathSciNetADSCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUSA
  2. 2.University of California Los AngelesLos AngelesUSA

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