Geodesic
Kazhdan–Lusztig correspondence for the representation category of the triplet $W$-algebra in logarithmic CFT
Teoretičeskaâ i matematičeskaâ fizika, Tome 148 (2006) no. 3, pp. 398-427 Cet article a éte moissonné depuis la source Math-Net.Ru

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To study the representation category of the triplet $W$-algebra $\boldsymbol{\mathcal{W}}(p)$ that is the symmetry of the $(1,p)$ logarithmic conformal field theory model, we propose the equivalent category $\EuScript{C}_p$ of finite-dimensional representations of the restricted quantum group $\overline{\EuScript{U}}_{\!\mathfrak{q}} s\ell(2)$ at $\mathfrak{q}=e^{{i\pi}/{p}}$. We fully describe the category $\EuScript{C}_p$ by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the $\boldsymbol{\mathcal{W}}(p)$- and $\overline{\EuScript{U}}_{\!\mathfrak{q}} s\ell(2)$-representation categories is conjectured for all $p\ge2$ and proved for $p=2$. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal $R$-matrix with the braiding matrix.
Mots-clés : Kazhdan–Lusztig correspondence, indecomposable representations.
Keywords: quantum groups, logarithmic conformal field theories 
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     title = {Kazhdan{\textendash}Lusztig correspondence for the~representation category of the~triplet $W$-algebra in logarithmic {CFT}},
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A. M. Gainutdinov; A. M. Semikhatov; I. Yu. Tipunin; B. L. Feigin. Kazhdan–Lusztig correspondence for the representation category of the triplet $W$-algebra in logarithmic CFT. Teoretičeskaâ i matematičeskaâ fizika, Tome 148 (2006) no. 3, pp. 398-427. http://geodesic.mathdoc.fr/item/TMF_2006_148_3_a4/

[1] D. Kazhdan, G. Lusztig, J. Amer. Math. Soc., 6 (1993), 905 ; 949 ; 7 (1994), 335 ; 383 | DOI | MR | Zbl | DOI | DOI | MR | Zbl | DOI | MR | Zbl

[2] M. Finkelberg, Geom. and Funct. Anal., 6 (1996), 249 ; V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, Walter de Gruyter, Berlin–New York, 1994 | DOI | MR | Zbl | MR | Zbl

[3] G. Moore, N. Seiberg, “Lectures on RCFT”, Superstrings'89, Proc. Spring School (Trieste, Italy, April 3–14, 1989), eds. M. Green, R. Iengo, S. Randjbar-Daemi, E. Sezgin, A. Strominger, World Scientific, Singapore, 1990, 263 | MR | Zbl

[4] J. Fuchs, I. Runkel, C. Schweigert, Nucl. Phys. B, 646 (2002), 353 ; hep-th/0204148 | DOI | MR | Zbl

[5] E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Math. Surv. Monographs, 28, AMS, Providence, RI, 2001 ; B. Bakalov, A. A. Kirillov, Lectures on Tensor Categories and Modular Functors, AMS, Providence, RI, 2001 ; J. Fuchs, I. Runkel, C. Schweigert, Nucl. Phys. B, 678 (2004), 511 ; hep-th/0306164 | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[6] J. Fuchs, I. Runkel, C. Schweigert, Ribbon categories and (unoriented) CFT: Frobenius algebras, automorphisms, reversions, math.CT/0511590 | MR

[7] M. Miyamoto, A theory of tensor products for vertex operator algebra satisfying $C_2$-cofiniteness, math.QA/0309350

[8] Y.-Z. Huang, J. Lepowsky, L. Zhang, A logarithmic generalization of tensor product theory for modules for a vertex operator algebra, math.QA/0311235 | MR

[9] J. Fuchs, On non-semisimple fusion rules and tensor categories, hep-th/0602051 | MR

[10] H. G. Kausch, Phys. Lett. B, 259 (1991), 448 | DOI | MR

[11] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Commun. Math. Phys., 265 (2006), 47 ; hep-th/0504093 | DOI | MR | Zbl

[12] J. Fuchs, S. Hwang, A. M. Semikhatov, I. Yu. Tipunin, Commun. Math. Phys., 247 (2004), 713 ; hep-th/0306274 | DOI | MR | Zbl

[13] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Studies Adv. Math., 36, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[14] M. A. I. Flohr, Int. J. Mod. Phys. A, 11 (1996), 4147 ; hep-th/9509166 | DOI | MR | Zbl

[15] T. Kerler, V. V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners, Lect. Notes in Math., 1765, Springer, Berlin, 2001 | MR | Zbl

[16] M. Flohr, M. R. Gaberdiel, J. Phys. A, 39 (2006), 1955 ; hep-th/0509075 | DOI | MR | Zbl

[17] V. Lyubashenko, Commun. Math. Phys., 172 (1995), 467 ; ; “Modular properties of ribbon abelian categories”, Proc. 2nd Gauss Symposium. Conference A: Mathematical and Theoretical Physics (Münich, Germany, August 2–7, 1993), eds. M. Behara, R. Fritish, R. G. Lintz, de Gruyter, Berlin, 1995, 529 ; ; J. Pure Applied Algebra, 98 (1995), 279 ; V. Lyubashenko, S. Majid, J. Algebra, 166 (1994), 506 hep-th/9405167hep-th/9405168 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[18] M. R. Gaberdiel, H. G. Kausch, Nucl. Phys. B, 477 (1996), 293 ; hep-th/9604026 | DOI | MR

[19] S. MacLane, Homology, Springer, Berlin, 1963 | MR | Zbl

[20] W. Crawley-Boevey, Lectures on representations of quivers, Lectures in Oxford in 1992, available at http://www.amsta.leeds.ac.uk/p̃mtwc/

[21] H. G. Kausch, Nucl. Phys. B, 583 (2000), 513 ; hep-th/0003029 | DOI | MR | Zbl

[22] M. R. Gaberdiel, H. G. Kausch, Phys. Lett. B, 386 (1996), 131 ; hep-th/9606050 | DOI | MR

[23] H. G. Kausch, Curiosities at $c=-2$, hep-th/9510149

[24] S. Arkhipov, R. Bezrukavnikov, V. Ginzburg, J. Am. Math. Soc., 17 (2004), 595 ; math.RT/0304173 | DOI | MR | Zbl

[25] Y.-Z. Huang, L. Kong, Full field algebras, math.QA/0511328 | MR

[26] P. Gabriel, Manuscripta Math., 6 (1972), 71 ; correction, ibid., 6 (1972), 309 ; И. Н. Бернштейн, И. М. Гельфанд, В. А. Пономарев, УМН, 28:2(170) (1973), 19 | DOI | MR | Zbl | DOI | MR | Zbl

[27] L. A. Nazarova, Izv. AN SSSR, ser. matem., 37 (1973), 752 | MR | Zbl

[28] V. Dlab, C. M. Ringel, Mem. Am. Math. Soc., 173, 1976 | MR | Zbl

[29] I. Frenkel, A. Malkin, M. Vybornov, Affine Lie algebras and tame quivers, math.RT/0005119 | MR

[30] L. Kronecker, Sitzungsber. Akad. Berlin, 1890 (1890), 1225 | Zbl