Commutative algebras in Fibonacci categories. (English) Zbl 1291.18010
Summary: By studying NIM-representations we show that the Fibonacci category and its tensor powers are completely anisotropic; that is, they do not have any non-trivial separable commutative ribbon algebras. As an application, we deduce that a chiral algebra with the representation category equivalent to a product of Fibonacci categories is maximal; that is, it is not a proper subalgebra of another chiral algebra. In particular the chiral algebras of the Yang-Lee model, the WZW models of \(G_{2}\) and \(F_{4}\) at level 1, as well as their tensor powers, are maximal.
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
References:
| [1] | Bakalov, B.; Kirillov, A., Lectures on Tensor Categories and Modular Functors, Univ. Lecture Ser., vol. 21 (2001), American Mathematical Society: American Mathematical Society Providence, RI, 221 pp · Zbl 0965.18002 |
| [2] | Belavin, A.; Polyakov, A.; Zamolodchikov, A., Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, 241, 2, 333-380 (1984) · Zbl 0661.17013 |
| [3] | Davydov, A.; Müger, M.; Nikshych, D.; Ostrik, V., The Witt group of non-degenerate braided fusion categories · Zbl 1271.18008 |
| [4] | Di Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal Field Theory, Grad. Texts Contemp. Phys. (1997), Springer-Verlag: Springer-Verlag New York, 890 pp · Zbl 0869.53052 |
| [5] | Dong, C.; Li, H.; Mason, G., Modular invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys., 214, 1, 1-56 (2000) · Zbl 1061.17025 |
| [6] | Drinfeld, V.; Gelaki, S.; Nikshych, D.; Ostrik, V., On braided fusion categories I, Selecta Math. (N.S.), 16, 1, 1-119 (2010) · Zbl 1201.18005 |
| [7] | Frenkel, I.; Zhu, Y., Vertex operator algebras associated to affine and Virasoro algebras, Duke Math. J., 66, 1, 123-168 (1992) · Zbl 0848.17032 |
| [8] | Fuchs, J.; Ganchev, A.; Vecsernyes, P., Rational Hopf algebras: polynomial equations, gauge fixing, and low-dimensional examples, Internat. J. Modern Phys. A, 10, 24, 3431-3476 (1995) · Zbl 1044.81628 |
| [9] | Fuchs, J.; Schellekens, A. N.; Schweigert, C., A matrix \(S\) for all simple current extensions, Nucl. Phys. B, 473, 1-2, 323-366 (1996) · Zbl 0921.17014 |
| [10] | Huang, Y.-Z., Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math., 10, Suppl. 1, 871-911 (2008) · Zbl 1169.17019 |
| [11] | Janelidze, G.; Kelly, G. M., A note on actions of a monoidal category, Theory Appl. Categ., 9, 61-91 (2001/2) · Zbl 1009.18005 |
| [12] | Joyal, A.; Street, R., Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra, 71, 1, 43-51 (1991) · Zbl 0726.18004 |
| [13] | Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 1, 20-78 (1993) · Zbl 0817.18007 |
| [14] | Kirillov, A.; Ostrik, V., On \(q\)-analog of McKay correspondence and ADE classification of \(\hat{sl}_2\) conformal field theories, Adv. Math., 171, 2, 183-227 (2002) · Zbl 1024.17013 |
| [15] | McCrudden, P., Categories of representations of coalgebroids, Adv. Math., 154, 2, 299-332 (2000) · Zbl 0968.18004 |
| [16] | Milas, A., Tensor product of vertex operator algebras |
| [17] | Moore, G.; Seiberg, N., Classical and quantum conformal field theory, Comm. Math. Phys., 123, 2, 177-254 (1989) · Zbl 0694.53074 |
| [18] | Moore, G.; Seiberg, N., Lectures on RCFT, (Superstrings 89. Superstrings 89, Trieste, 1989 (1990), World Sci. Publ.), 1-129 · Zbl 0985.81739 |
| [19] | Müger, M., From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra, 180, 1-2, 159-219 (2003) · Zbl 1033.18003 |
| [20] | Müger, M., On the structure of modular categories, Proc. Lond. Math. Soc., 87, 2, 291-308 (2003) · Zbl 1037.18005 |
| [21] | Ostrik, V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups, 8, 2, 177-206 (2003) · Zbl 1044.18004 |
| [22] | Ostrik, V., Fusion categories of rank 2, Math. Res. Lett., 10, 2-3, 177-183 (2003) · Zbl 1040.18003 |
| [23] | Pareigis, B., On braiding and dyslexia, J. Algebra, 171, 2, 413-425 (1995) · Zbl 0816.18003 |
| [24] | Quillen, D., Higher algebraic \(K\)-theory. I, (Algebraic \(K\)-theory, I: Higher \(K\)-theories. Algebraic \(K\)-theory, I: Higher \(K\)-theories, Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972. Algebraic \(K\)-theory, I: Higher \(K\)-theories. Algebraic \(K\)-theory, I: Higher \(K\)-theories, Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972, Lecture Notes in Math., vol. 341 (1973)), 85-147 · Zbl 0292.18004 |
| [25] | Schellekens, A., Meromorphic \(c = 24\) conformal field theories, Comm. Math. Phys., 153, 1, 159-185 (1993) · Zbl 0782.17014 |
| [26] | Turaev, V., Quantum Invariants of Knots and 3-Manifolds, de Gruyter Stud. Math., vol. 18 (1994), Walter de Gruyter: Walter de Gruyter Berlin, 588 pp · Zbl 0812.57003 |
| [27] | Wang, W., Rationality of Virasoro vertex operator algebras, Int. Math. Res. Not. IMRN, 71, 7, 197-211 (1993) · Zbl 0791.17029 |
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