Irreducible weight modules over Witt algebras

Guo, Xiangqian; Zhao, Kaiming

doi:10.1090/S0002-9939-2010-10679-2

Irreducible weight modules over Witt algebras

Authors: Xiangqian Guo and Kaiming Zhao
Journal: Proc. Amer. Math. Soc. 139 (2011), 2367-2373
MSC (2010): Primary 17B10, 17B20, 17B65, 17B66, 17B68
DOI: https://doi.org/10.1090/S0002-9939-2010-10679-2
Published electronically: December 9, 2010
MathSciNet review: 2784801
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1986, Shen defined a class of modules over the Witt algebra from irreducible modules over the general linear Lie algebra $\mathfrak{gl}_d$ , which were also given by Larsson in 1992. In 1996, Eswara Rao determined the irreducibility of these modules. In this paper, we use simpler methods to give a short and straightforward proof to the results of Eswara Rao.

[AABGP] Bruce N. Allison, Saeid Azam, Stephen Berman, Yun Gao, and Arturo Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. 126 (1997), no. 603, x+122. MR 1376741, https://doi.org/10.1090/memo/0603
[B] Yuly Billig, A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not. , posted on (2006), Art. ID 68395, 46. MR 2272091, https://doi.org/10.1155/IMRN/2006/68395
[BMZ] Yuly Billig, Alexander Molev, and Ruibin Zhang, Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus, Adv. Math. 218 (2008), no. 6, 1972–2004. MR 2431666, https://doi.org/10.1016/j.aim.2008.03.026
[BZ] Yuly Billig and Kaiming Zhao, Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004), no. 1-2, 23–42. MR 2048305, https://doi.org/10.1016/j.jpaa.2003.12.004
[E1] S. Eswara Rao, Irreducible representations of the Lie-algebra of the diffeomorphisms of a 𝑑-dimensional torus, J. Algebra 182 (1996), no. 2, 401–421. MR 1391590, https://doi.org/10.1006/jabr.1996.0177
[E2] S. Eswara Rao, Irreducible representations for toroidal Lie-algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 102–117. MR 2163403, https://doi.org/10.1016/j.jpaa.2005.01.011
[E3] S. Eswara Rao, Partial classification of modules for Lie algebra of diffeomorphisms of 𝑑-dimensional torus, J. Math. Phys. 45 (2004), no. 8, 3322–3333. MR 2077512, https://doi.org/10.1063/1.1769104
[EJ] S. Eswara Rao and Cuipo Jiang, Classification of irreducible integrable representations for the full toroidal Lie algebras, J. Pure Appl. Algebra 200 (2005), no. 1-2, 71–85. MR 2142351, https://doi.org/10.1016/j.jpaa.2004.12.051
[KR] V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1021978
[KM] Oleksandr Khomenko and Volodymyr Mazorchuk, Generalized Verma modules induced from 𝑠𝑙(2,ℂ) and associated Verma modules, J. Algebra 242 (2001), no. 2, 561–576. MR 1848960, https://doi.org/10.1006/jabr.2001.8815
[L1] T. A. Larsson, Multi-dimensional Virasoro algebra, Phys. Lett. B 231 (1989), no. 1-2, 94–96. MR 1023577, https://doi.org/10.1016/0370-2693(89)90119-6
[L2] T. A. Larsson, Central and noncentral extensions of multi-graded Lie algebras, J. Phys. A 25 (1992), no. 5, 1177–1184. MR 1154861
[L3] T. A. Larsson, Conformal fields: a class of representations of 𝑉𝑒𝑐𝑡(𝑁), Internat. J. Modern Phys. A 7 (1992), no. 26, 6493–6508. MR 1186508, https://doi.org/10.1142/S0217751X92002970
[L4] T. A. Larsson, Lowest-energy representations of non-centrally extended diffeomorphism algebras, Comm. Math. Phys. 201 (1999), no. 2, 461–470. MR 1682285, https://doi.org/10.1007/s002200050563
[L5] T. A. Larsson, Extended diffeomorphism algebras and trajectories in jet space, Comm. Math. Phys. 214 (2000), no. 2, 469–491. MR 1796031, https://doi.org/10.1007/s002200000280
[M1] Olivier Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225–234. MR 1144422, https://doi.org/10.1007/BF01231888
[M2] Olivier Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537–592 (English, with English and French summaries). MR 1775361
[MZ] V. Mazorchuk and K. Zhao, Supports of weight modules over Witt algebras, Proc. R. Soc. Edinb. Sect. A-Math., in press.
[Sh] Guang Yu Shen, Graded modules of graded Lie algebras of Cartan type. I. Mixed products of modules, Sci. Sinica Ser. A 29 (1986), no. 6, 570–581. MR 862418
[Z] Kaiming Zhao, Weight modules over generalized Witt algebras with 1-dimensional weight spaces, Forum Math. 16 (2004), no. 5, 725–748. MR 2096685, https://doi.org/10.1515/form.2004.034