Let $ \widetilde{\mathfrak{sv}} $ be the extended Schrödinger-Virasoro Lie algebra. In this paper, we consider the skew-symmetric biderivations of the extended Schrödinger-Virasoro Lie algebra. We prove that all biderivations of $ \widetilde{\mathfrak{sv}} $ are inner. Based on this result, we show that all linear commuting maps on $ \widetilde{\mathfrak{sv}} $, which have the form $ \psi(x) = \lambda x $, are standard.
Citation: Zhongxian Huang. Biderivations of the extended Schrödinger-Virasoro Lie algebra[J]. AIMS Mathematics, 2023, 8(12): 28808-28817. doi: 10.3934/math.20231476
Let $ \widetilde{\mathfrak{sv}} $ be the extended Schrödinger-Virasoro Lie algebra. In this paper, we consider the skew-symmetric biderivations of the extended Schrödinger-Virasoro Lie algebra. We prove that all biderivations of $ \widetilde{\mathfrak{sv}} $ are inner. Based on this result, we show that all linear commuting maps on $ \widetilde{\mathfrak{sv}} $, which have the form $ \psi(x) = \lambda x $, are standard.
[1] |
J. Unterberger, On vertex algebra representations of the Schrödinger-Virasoro Lie algebra, Nucl. Phys. B, 823 (2009), 320–371. https://doi.org/10.1016/j.nuclphysb.2009.06.018 doi: 10.1016/j.nuclphysb.2009.06.018
![]() |
[2] |
M. Henkel, Schrödinger invariance and strongly anisotropic critical systems, J. Stat. Phys., 75 (1994), 1023–1061. https://doi.org/10.1007/BF02186756 doi: 10.1007/BF02186756
![]() |
[3] |
J. Li, Y. Su, Representations of the Schrödinger-Virasoro algebras, J. Math. Phys., 49 (2008), 053512. https://doi.org/10.1063/1.2924216 doi: 10.1063/1.2924216
![]() |
[4] |
Y. Su, L. Yuan, Schrödinger-Virasoro Lie conformal algebra, J. Math. Phys., 54 (2013), 053503. https://doi.org/10.1063/1.4803029 doi: 10.1063/1.4803029
![]() |
[5] |
Y. Pei, C. Bai, Novikov algebras and Schrödinger-Virasoro Lie algebras, J. Phys. A: Math. Theor., 44 (2011), 045201. https://doi.org/10.1088/1751-8113/44/4/045201 doi: 10.1088/1751-8113/44/4/045201
![]() |
[6] |
D. Wang, X. Yu, Biderivation and linear commuting maps on the Schrödinger-Virasoro Lie algebra, Commun. Algebra, 41 (2013), 2166–2173. https://doi.org/10.1080/00927872.2012.654551 doi: 10.1080/00927872.2012.654551
![]() |
[7] |
J. Wen, Z. Xu, Y. Hong, Non-weight modules over a Schrödinger-Virasoro type algebra, Linear Multilinear A., 2023 (2023), 2176419. https://doi.org/10.1080/03081087.2023.2176419 doi: 10.1080/03081087.2023.2176419
![]() |
[8] |
P. Xu, X. Tang, Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra, Linear Multilinear A., 29 (2021), 2771–2789. https://doi.org/10.3934/era.2021013 doi: 10.3934/era.2021013
![]() |
[9] |
S. Gao, C. Jiang, Y. Pei, Structure of the extended Schrödinger-Virasoro Lie algebra formula, Algebr. Colloq., 16 (2009), 549–566. https://doi.org/10.1142/S1005386709000522 doi: 10.1142/S1005386709000522
![]() |
[10] |
L. Yuan, Y. Wu, Y. Xu, Lie bialgebra structures on the extended Schrödinger-Virasoro Lie algebra, Algebr. Colloq., 18 (2011), 709–720. https://doi.org/10.1142/S1005386711000563 doi: 10.1142/S1005386711000563
![]() |
[11] |
Y. Zhong, X. Tang, n-Derivations of the extended Schrödinger-Virasoro Lie algebra, Algebr. Colloq., 28 (2021), 105–118. https://doi.org/10.1142/S1005386721000109 doi: 10.1142/S1005386721000109
![]() |
[12] |
D. Benkovic, Biderivations of triangular algebras, Linear Algebra Appl., 431 (2009), 1587–1602. https://doi.org/10.1016/j.laa.2009.05.029 doi: 10.1016/j.laa.2009.05.029
![]() |
[13] |
D. Wang, X. Yu, Z. Chen, Biderivations of the parabolic subalgebras of simple Lie algebras, Commun. Algebra, 39 (2011), 4097–4104. https://doi.org/10.1080/00927872.2010.517820 doi: 10.1080/00927872.2010.517820
![]() |
[14] |
M. Abdaoui, Skew-symmetric biderivations and linear commuting maps on the conformal Galilei algebra, Commun. Algebra, 51 (2023), 1663–1672. https://doi.org/10.1080/00927872.2022.2140348 doi: 10.1080/00927872.2022.2140348
![]() |
[15] |
Z. Chen, Y. Wang, Skew-symmetric biderivations and linear commuting maps of Kac-Moody algebras, Linear Multilinear A., 71 (2023), 867–874. https://doi.org/10.1080/03081087.2022.2044747 doi: 10.1080/03081087.2022.2044747
![]() |
[16] | G. Fan, Y. Su, K. Xu, Linear commuting maps and skew-symmertric biderivations of the deformative Schrödinger-Virasoro Lie algebras, arXiv: 1611.05718. https://doi.org/10.48550/arXiv.1611.05718 |