September  2021, 29(4): 2673-2685. doi: 10.3934/era.2021008

Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra

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

Received  July 2020 Revised  December 2020 Published  September 2021 Early access  January 2021

Fund Project: Partially supported by NSFC (No.11901224) and NSF of Hubei Province (No.2019CFB160)

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $ V_{\mathcal{L}}(\ell_{123},0) $. Then, for any integer $ t>1 $, we introduce a new Lie algebra $ \mathcal{L}_{t} $, and show that $ \sigma_{t} $-twisted $ V_{\mathcal{L}}(\ell_{123},0) $($ \ell_{2} = 0 $)-modules are in one-to-one correspondence with restricted $ \mathcal{L}_{t} $-modules of level $ \ell_{13} $, where $ \sigma_{t} $ is an order $ t $ automorphism of $ V_{\mathcal{L}}(\ell_{123},0) $. At the end, we give a complete list of irreducible $ \sigma_{t} $-twisted $ V_{\mathcal{L}}(\ell_{123},0) $($ \ell_{2} = 0 $)-modules.

Citation: Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008
References:
[1]

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R. Shen and C. Jiang, The derivation algebra and automorphism group of the twisted Heisenberg-Virasoro algebra, Comm. Algebra, 34 (2006), 2547-2558.  doi: 10.1080/00927870600651257.  Google Scholar

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W. Wang, Rationality of virasoro vertex operator algebras, Internat. Math. Res. Notices, (1993), 197–211. doi: 10.1155/S1073792893000212.  Google Scholar

show all references

References:
[1]

D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra, 219 (2015), 4322–4342. doi: 10.1016/j.jpaa.2015.02.019.  Google Scholar

[2]

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, 26 pp. doi: 10.1142/S0219199718500086.  Google Scholar

[3]

E. ArbarelloC. De ConciniV. G. Kac and C. Procesi, Moduli spaces of curves and representation theory, Comm. Math. Phys., 117 (1988), 1-36.  doi: 10.1007/BF01228409.  Google Scholar

[4]

Y. Billig, Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canad. Math. Bull., 46 (2003), 529-537.  doi: 10.4153/CMB-2003-050-8.  Google Scholar

[5]

Y. Billig, A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not., (2006), Art. ID. 68395, 46pp. doi: 10.1155/IMRN/2006/68395.  Google Scholar

[6]

C. Dong and K. Nagatomo, Classification of irreducible modules for the vertex operator algebra $M(1)^+$, J. Algebra, 216 (1999), 384-404.  doi: 10.1006/jabr.1998.7784.  Google Scholar

[7]

I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc., 104 (1993), no. 494, ⅷ+64 pp. doi: 10.1090/memo/0494.  Google Scholar

[8]

I. B. Frenkel and A. M. Zeitlin, Quantum group $GL_{q}(2)$ and quantum Laplace operator via semi-infinite cohomology, J. Noncommut. Geom., 7 (2013), 1007-1026.  doi: 10.4171/JNCG/142.  Google Scholar

[9]

I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66 (1992), 123-168.  doi: 10.1215/S0012-7094-92-06604-X.  Google Scholar

[10]

H. GuoH. LiS. Tan and Q. Wang, $q$-Virasoro algebra and vertex algebras, J. Pure Appl. Algebra, 219 (2015), 1258-1277.  doi: 10.1016/j.jpaa.2014.06.004.  Google Scholar

[11]

H. Guo and Q. Wang, Associating vertex algebras with the unitary Lie algebra, J. Algebra, 424 (2015), 126-146.  doi: 10.1016/j.jalgebra.2014.11.006.  Google Scholar

[12]

H. Guo and Q. Wang, Twisted Heisenberg-Virasoro vertex operator algebra, Glas. Mat. Ser. Ⅲ, 54 (2019), 369-407.   Google Scholar

[13]

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and their Representations, Progress in Mathematics, Vol. 227, Birkhäuser, Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-0-8176-8186-9.  Google Scholar

[14]

H.-S. Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, In: Moonshine, the Monster, and related topics, pp. 203–236, Contemp. Math. 193, Amer. Math. Soc., Providence, RI, 1996. doi: 10.1090/conm/193/02373.  Google Scholar

[15]

R. Shen and C. Jiang, The derivation algebra and automorphism group of the twisted Heisenberg-Virasoro algebra, Comm. Algebra, 34 (2006), 2547-2558.  doi: 10.1080/00927870600651257.  Google Scholar

[16]

W. Wang, Rationality of virasoro vertex operator algebras, Internat. Math. Res. Notices, (1993), 197–211. doi: 10.1155/S1073792893000212.  Google Scholar

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