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Vertex algebraic intertwining operators among generalized Verma modules for $ \widehat{\mathfrak{sl}(2,\mathbb{C})}$

Authors: Robert McRae and Jinwei Yang
Journal: Trans. Amer. Math. Soc. 370 (2018), 2351-2390
MSC (2010): Primary 17B67, 17B69
Published electronically: November 30, 2017
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Abstract: We construct vertex algebraic intertwining operators among certain generalized Verma modules for $ \widehat {\mathfrak{sl}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $ \widehat {\mathfrak{sl}(2,\mathbb{C})}$-modules. We prove this irreducibility using the composition factor multiplicities of irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras of rank $ 2$, given by Rocha-Caridi and Wallach.

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  • [C] Luis Casian, Kazhdan-Lusztig multiplicity formulas for Kac-Moody algebras, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 333-337 (English, with French summary). MR 1046507
  • [DGK] Vinay V. Deodhar, Ofer Gabber, and Victor Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), no. 1, 92-116. MR 663417,
  • [FHL] Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494,
  • [FLM] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
  • [FZ] Igor B. Frenkel and Yongchang Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123-168. MR 1159433,
  • [GL] Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37-76. MR 0414645,
  • [H1] Yi-Zhi Huang, A theory of tensor products for module categories for a vertex operator algebra. IV, J. Pure Appl. Algebra 100 (1995), no. 1-3, 173-216. MR 1344849,
  • [H2] Yi-Zhi Huang, Differential equations, duality and modular invariance, Commun. Contemp. Math. 7 (2005), no. 5, 649-706. MR 2175093,
  • [H3] Yi-Zhi Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008), no. 1, 103-154. MR 2387861,
  • [H4] Yi-Zhi Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), no. suppl. 1, 871-911. MR 2468370,
  • [HL] Yi-Zhi Huang and James Lepowsky, Tensor categories and the mathematics of rational and logarithmic conformal field theory, J. Phys. A 46 (2013), no. 49, 494009, 21. MR 3146015,
  • [HLZ1] Y.-Z. Huang, J, Lepowsky, and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, Conformal Field Theories and Tensor Categories, Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, ed. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2, Springer, New York, 2014, 169-248.
  • [HLZ2] Y.-Z. Huang, J. Lepowsky, and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms, arXiv:1012.4202.
  • [Hu] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
  • [K] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [Ka] Masaki Kashiwara, Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 407-433. MR 1106905
  • [KaT] Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II. Intersection cohomologies of Schubert varieties, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 159-195. MR 1103590
  • [KK] V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97-108. MR 547842,
  • [KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412,
  • [Le1] J. Lepowsky, Lectures on Kac-Moody Lie algebras, mimeographed notes, Paris: Université de Paris VI 1978.
  • [Le2] J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), no. 2, 470-495. MR 0463360,
  • [LL] James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933
  • [Li1] Hai Sheng Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 279-297. MR 1303287,
  • [Li2] Haisheng Li, An analogue of the Hom functor and a generalized nuclear democracy theorem, Duke Math. J. 93 (1998), no. 1, 73-114. MR 1620083,
  • [Li3] Haisheng Li, Determining fusion rules by $ A(V)$-modules and bimodules, J. Algebra 212 (1999), no. 2, 515-556. MR 1676853,
  • [MP] Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1323858
  • [RW1] Alvany Rocha-Caridi and Nolan R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), no. 2, 151-177. MR 661694,
  • [RW2] Alvany Rocha-Caridi and Nolan R. Wallach, Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas, Trans. Amer. Math. Soc. 277 (1983), no. 1, 133-162. MR 690045,
  • [TK] Akihiro Tsuchiya and Yukihiro Kanie, Vertex operators in conformal field theory on $ {\bf P}^1$ and monodromy representations of braid group, Conformal field theory and solvable lattice models (Kyoto, 1986) Adv. Stud. Pure Math., vol. 16, Academic Press, Boston, MA, 1988, pp. 297-372. MR 972998
  • [V] Daya-Nand Verma, Structure of certain induced representations of complex semisimple Lie algebras, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)-Yale University. MR 2615829
  • [Z] Yongchang Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237-302. MR 1317233,

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Additional Information

Robert McRae
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100084, People’s Republic of China
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Jinwei Yang
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematics, Yale University, New Haven, Connecticut 06520

Received by editor(s): December 14, 2015
Received by editor(s) in revised form: June 23, 2016
Published electronically: November 30, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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