Abstract
We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.
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Behrend R.E., Pearce P.A., Petkova V.B., Zuber J.-B.: Boundary conditions in rational conformal field theories. Nucl. Phys. B 579, 707–773 (2000)
Bion-Nadal, J.: An example of a subfactor of the hyperfinite II 1 factor whose principal graph invariant is the Coxeter graph E 6. In: Current topics in operator algebras (Nara, 1990), River Edge, NJ: World Sci. Publ., 1991, pp. 104–113
Böckenhauer, J.: Lecture at Warwick Workshop on Modular Invariants, Operator Algebras and Quotient Singularities, September 1999
Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. I. Commun. Math. Phys. 197, 361–386 (1998)
Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999)
Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. III. Commun. Math. Phys. 205, 183–228 (1999)
Böckenhauer J., Evans D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267–289 (2000)
Böckenhauer, J., Evans, D.E.: Modular invariants and subfactors. In: Mathematical physics in mathematics and physics (Siena, 2000), Fields Inst. Commun. 30, 11–37, Providence, RI, Amer. Math. Soc., 2001, pp. 11–37
Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors. In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 95–131
Böckenhauer J., Evans D.E., Kawahigashi Y.: On α-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)
Böckenhauer J., Evans D.E., Kawahigashi Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000)
Bocklandt R.: Graded Calabi Yau algebras of dimension 3. J. Pure Appl. Algebra 212, 14–32 (2008)
Brenner S., Butler M.C.R., King A.D.: Periodic algebras which are almost Koszul. Algebr. Represent. Theory 5, 331–367 (2002)
Broomhead, N.: Dimer models and Calabi-Yau algebras. PhD thesis, University of Bath, 2008
Cappelli A., Itzykson C., Zuber J.-B.: The A−D−E classification of minimal and A (1)1 conformal invariant theories. Commun. Math. Phys. 113, 1–26 (1987)
Cooper, B.: Almost Koszul Duality and Rational Conformal Field Theory. PhD thesis, University of Bath, 2007
Crawley-Boevey W., Holland M.P.: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92, 605–635 (1998)
Di Francesco P., Zuber J.-B.: SU(N) lattice integrable models associated with graphs. Nucl. Phys. B 338, 602–646 (1990)
Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras. I, II. J. Algebra 205, 391–412, 413–434 (1998)
Erdmann, K., Snashall, N.: Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology. In: Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc. 24, Providence, RI: Amer. Math. Soc., 1998, pp. 183–193
Etingof P., Ostrik V.: Module categories over representations of SL q (2) and graphs. Math. Res. Lett. 11, 103–114 (2004)
Evans, D.E.: Critical phenomena, modular invariants and operator algebras. In: Operator algebras and mathematical physics (Constanţa, 2001), Bucharest: Theta, 2003, pp. 89–113
Evans D.E., Kawahigashi Y.: Orbifold subfactors from Hecke algebras. Commun. Math. Phys. 165, 445–484 (1994)
Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras, Oxford Mathematical Monographs. New York: The Clarendon Press/Oxford University Press, 1998
Evans D.E., Pugh M.: Ocneanu Cells and Boltzmann Weights for the \({SU(3)\,\mathcal{ADE}}\) Graphs. Münster J. Math. 2, 95–142 (2009)
Evans D.E., Pugh M.: SU(3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU(3) modular invariants. Rev. Math. Phys. 21, 877–928 (2009)
Evans, D.E., Pugh, M.: A 2-Planar Algebras I, Quantum Topol., 1, 321–377 (2010) Updated version after pub. at http://arxiv.org/abs/0906.9225v6 [math.OA], 2011
Evans D.E., Pugh M.: A 2-Planar Algebras II: Planar Modules. J. Funct. Anal, 261, 1923–1954 (2011)
Evans D.E., Pugh M.: Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory. Commun. Math. Phys. 295, 363–413 (2010)
Evans D.E., Pugh M.: Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3). Commun. Math. Phys. 301, 771–809 (2011)
Evans, D.E., Pugh, M.: The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants. http://arxiv.org/abs/1008.1003v2 [math.OA], 2011
Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance, Rev. Math. Phys. Special issue, 113–157 (1992)
Fröhlich J., Gabbiani F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251–353 (1990)
Fuchs J., Runkel I., Schweigert C.: Twenty five years of two-dimensional rational conformal field theory. J. Math. Phys. 51, 015210 (2010)
Gannon T.: The classification of affine SU(3) modular invariant partition functions. Commun. Math. Phys. 161, 233–263 (1994)
Gannon, T.: Private communication, 2001
Gel’fand I.M., Ponomarev V.A.: Model algebras and representations of graphs. Funk. Anal. i Pril. 13, 1–12 (1979)
Ginzburg, V.: Calabi-Yau algebras. http://arxiv.org/abs/math/0612139v3 [math.AG], 2007
Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras MSRI Publications, 14, New York: Springer-Verlag, 1989
Goodman F.M. Wenzl H.: Ideals in the Temperley-Lieb Category. Appendix to Freedman, Michael H., A magnetic model with a possible Chern-Simons phase. Commun. Math. Phys. 234, 129–183 (2003)
Graves, T.: Representations of affine truncations of representation involutive-semirings of Lie algebras and root systems of higher type, MSc thesis, University of Alberta, 2010
Izumi M.: Application of fusion rules to classification of subfactors. Publ. Res. Inst. Math. Sci. 27, 953–994 (1991)
Izumi M.: On flatness of the Coxeter graph E 8. Pacific J. Math. 166, 305–327 (1994)
Izumi M.: Subalgebras of infinite C*-algebras with finite Watatani indices. II. Cuntz-Krieger algebras. Duke Math. J. 91, 409–461 (1998)
Jimbo M.: A q-analogue of \({U(\mathfrak{gl}(N+1))}\), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)
Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Jones, V.F.R.: The planar algebra of a bipartite graph. In: Knots in Hellas ’98 (Delphi), Ser. Knots Everything 24, River Edge, NJ: World Sci. Publ., 2000, pp. 94–117
Kac, V.G.: Infinite-dimensional Lie algebras. Third Edition, Cambridge: Cambridge University Press, 1990
Kauffman L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)
Kawahigashi Y.: On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors. J. Funct. Anal. 127, 63–107 (1995)
Kirillov A. Jr, Ostrik V.: On a q-analogue of the McKay correspondence and the ADE classification of \({\mathfrak{sl}_2}\) conformal field theories. Adv. Math. 171, 183–227 (2002)
Kosaki H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986)
Kuperberg G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180, 109–151 (1996)
Longo R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130, 285–309 (1990)
Malkin A., Ostrik V., Vybornov M.: Quiver varieties and Lusztig’s algebra. Adv. Math. 203, 514–536 (2006)
Morrison S., Peters E., Snyder N.: Skein Theory for the D 2n Planar Algebras. J. Pure Appl. Algebra 214, 117–139 (2010)
Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser. 136, 119–172, Cambridge: Cambridge Univ. Press, 1988, pp. 119–172
Ocneanu, A.: Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors. (Notes recorded by S. Goto). In: Lectures on operator theory, (ed. B. V. Rajarama Bhat et al.), The Fields Institute Monographs, Providence, R.I.: Amer. Math. Soc., 2000, pp. 243–323
Ocneanu, A.: Higher Coxeter Systems (2000). Talk given at MSRI. http://www.msti.org/web/msri/online-videos/-/video/showsemester/pre2004
Ocneanu, A.: The classification of subgroups of quantum SU(N). In Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 133–159
Ohtsuki T., Yamada S.: Quantum SU(3) invariant of 3-manifolds via linear skein theory. J. Knot Theory Ramifications 6, 373–404 (1997)
Ostrik V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)
Reid, M.: La correspondance de McKay, Astérisque (2002), 53–72. Séminaire Bourbaki, Vol. 1999/2000, exp. no. 867, 53–72, available at http://archive.numdam.org/ARCHIVE/SB/SB_1999-2000_42_/SB_1999-2000_42_53/SB_1999-2000_42_53_O.pdf
Suciu, L.C.: The SU(3) Wire Model. PhD thesis, The Pennsylvania State University, 1997
Turaev, V.G.: Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics, 18, Berlin: Walter de Gruyter & Co. 1994
van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, Berlin: Springer, 2004, pp. 749–770
Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133, 467–538 (1998)
Wenzl H.: On sequences of projections. C. R. Math. Rep. Acad. Sci. Canada 9, 5–9 (1987)
Wenzl H.: Hecke algebras of type A n and subfactors. Invent. Math. 92, 349–383 (1988)
Xu F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 349–403 (1998)
Yamagami, S.: A categorical and diagrammatical approach to Temperley-Lieb algebras. http://arxiv.org/abs/math/0405267v2 [math.QA]
Yamagata, K.: Frobenius algebras. In: Handbook of algebra, Vol. 1, Amsterdam: Elsevier, 1996, pp. 841–887
Zuber, J.-B.: CFT, BCFT, ADE and all that. In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 233–266
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Evans, D.E., Pugh, M. The Nakayama Automorphism of the Almost Calabi-Yau Algebras Associated to SU(3) Modular Invariants. Commun. Math. Phys. 312, 179–222 (2012). https://doi.org/10.1007/s00220-011-1389-4
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DOI: https://doi.org/10.1007/s00220-011-1389-4