Glasnik Matematicki, Vol. 54, No. 2 (2019), 369-407.

TWISTED HEISENBERG-VIRASORO VERTEX OPERATOR ALGEBRA

Hongyan Guo and Qing Wang

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Abstract.   In this paper, we study a new kind of vertex operator algebras related to twisted Heisenberg-Virasoro algebra. We showed that there exist one-to-one correspondences between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study in detail the structures of the twisted Heisenberg-Virasoro vertex operator algebra and give a characterization of it as a tensor product of two well-known vertex operator algebras.

2010 Mathematics Subject Classification.   17B69

Key words and phrases.   Twisted Heisenberg-Virasoro algebra, vertex operator algebra, modules

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.2.06

References:

  1. T. Abe, G. Buhl and C. Dong, Rationality, regularity, and C2-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), 3391-3402.
    MathSciNet     CrossRef

  2. E. Arbarello, C. De Concini, V. Kac and C. Procesi, Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988), 1-36.
    MathSciNet     CrossRef

  3. D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications, J. Pure and Appl. Algebra 219 (2015), 4322-4342.
    MathSciNet     CrossRef

  4. D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg-Virasoro algebra at level zero, Commun. Contemp. Math. 21 (2019), 1850008, 26pp.
    MathSciNet     CrossRef

  5. Y. Billig, Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canad. Math. Bull. 46 (2003), 529-537.
    MathSciNet     CrossRef

  6. Y. Billig, A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not. 2006, Art. ID. 68395, 46pp.
    MathSciNet     CrossRef

  7. S. Berman, Y. Billig and J. Szmigielski, Vertex operator algebras and the representation theory of toroidal algebras, Contemporary Math. 297, Amer. Math. Soc., Providence, 2002, 1-26.
    MathSciNet     CrossRef

  8. C. Dong and J. Lepowsky, Generalized vertex algebras and relative vertex operators, Birkhauser Boston, Inc., Boston, 1993.
    MathSciNet     CrossRef

  9. C. Dong and X.Lin, Unitary vertex operator algebras, J. Algebra 397 (2014), 252-277.
    MathSciNet     CrossRef

  10. C. Dong and Q. Wang, The structure of parafermion vertex operator algebras, Commun. Math. Phys. 299 (2010), 783-792.
    MathSciNet     CrossRef

  11. C. Dong, H. Li and G. Mason, Regularity of rational vertex operator algebra, Adv. Math. 312 (1997), 148-166.
    MathSciNet     CrossRef

  12. C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571-600.
    MathSciNet     CrossRef

  13. I. Frenkel, Y. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, Viii+64 pp.
    MathSciNet     CrossRef

  14. I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster, Academic Press, Inc., Boston, 1988.
    MathSciNet    

  15. I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
    MathSciNet     CrossRef

  16. M. Golenishcheva-Kutuzova and V. Kac, γ-conformal algebras, J. Math. Phys. 39 (1998), 2290-2305.
    MathSciNet     CrossRef

  17. P. Goddard, A. Kent and D. Olive, Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), 88-92.
    MathSciNet     CrossRef

  18. H. Guo, H. Li, S. Tan and Q. Wang, q-Virasoro algebra and vertex algebras, J. Pure Appl. Algebra 219 (2015), 1258-1277.
    MathSciNet     CrossRef

  19. H. Guo and Q. Wang, Associating vertex algebras to the unitary Lie algebra, J. Algebra 424 (2015), 126-146.
    MathSciNet     CrossRef

  20. C. Jiang and H. Li, Associating quantum vertex algebras to Lie algebra 𝔤𝔩∞, J. Algebra 399 (2014), 1086-1106.
    MathSciNet     CrossRef

  21. C. Jiang and Z. Lin, The commutant of L(n,0) in the vertex operator algebra L(1,0)⊗ n, Adv. Math. 301 (2016), 227-257.
    MathSciNet     CrossRef

  22. V. Kac and D. Peterson, Infinite-dimensional Lie algebra, theta functions and modular forms, Adv. Math. 53 (1984), 125-264.
    MathSciNet     CrossRef

  23. B. Lian and A. Linshaw, Howe pairs in the theory of vertex algebras, J. Algebra 317 (2007), 111-152.
    MathSciNet     CrossRef

  24. J. Lepowsky and H. Li, Introduction to vertex operator algebras and their representations, Birkhäuser, Inc., Boston, 2004.
    MathSciNet     CrossRef

  25. H. Li, A new construction of vertex algebras and quasi modules for vertex algebras, Adv. Math. 202 (2006), 232-286.
    MathSciNet     CrossRef

  26. H. Li, On certain generalizations of twisted affine Lie algebras and quasimodules for γ-vertex algebras, J. Pure Appl. Algebra 209 (2007), 853-871.
    MathSciNet     CrossRef

  27. H. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), 143-195.
    MathSciNet     CrossRef

  28. H. Li, φ-coordinated quasi-modules for quantum vertex algebras, Comm. Math. Phys. 308 (2011), 703-741.
    MathSciNet     CrossRef

  29. H. Li, G-equivariant φ-coordinated quasi-modules for quantum vertex algebras, J. Math. Phys. 54 (2013), 051704, 26 pp.
    MathSciNet     CrossRef

  30. H. Li, Nonlocal vertex algebras generated by formal vertex operators, Selecta Math. (N.S.) 11 (2005), 349-397.
    MathSciNet     CrossRef

  31. H. Li, Some finiteness properties of regular vertex operator algebras, J. Algebra 212 (1999), 495-514.
    MathSciNet     CrossRef

  32. H. Li, S. Tan and Q. Wang, Toroidal vertex algebras and their modules, J. Algebra 365 (2012) 50-82.
    MathSciNet     CrossRef

  33. M. Xue, W. Lin and S. Tan, Central extension, derivations and automorphism group for Lie algebras arising from 2-dimensional torus, J. Lie Theory 16 (2006), 139-153.
    MathSciNet    

  34. V. Kac and Rehana, Bombay lectures on highest weight representation of infinite dimensional Lie algebras, World Scientific Publishing Co., Inc., Teaneck, 1987.
    MathSciNet    

  35. W. Wang, Rationality of Virasoro vertex operator algebras, Internat. Math. Res. Notices 1993, 197-211.
    MathSciNet     CrossRef

  36. Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc 9 (1996), 237-302.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page