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  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, doi: 10.3934/era.2021008
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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

[16]

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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