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Whittaker modules for the derivation Lie algebra of torus with two variables. (English) Zbl 1348.17008

Summary: Let \(\mathcal L\) be the derivation Lie algebra of \(\mathbb{C}\left[t_1^{ \pm 1},t_2^{ \pm 1} \right]\). Given a triangle decomposition \(\mathcal L=\mathcal L^{+} \oplus \mathfrak h \oplus \mathcal L^-\), we define a nonsingular Lie algebra homomorphism \(\psi: \mathcal L^{+} \to \mathbb{C}\) and the universal Whittaker \(\mathcal L\)-module \(W_{\psi}\) of type \(\psi\). We obtain all Whittaker vectors and submodules of \(W_{\psi}\). Moreover, all simple Whittaker \(\mathcal L\)-modules of type \(\psi\) are determined.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65 Infinite-dimensional Lie (super)algebras
17B68 Virasoro and related algebras
17B81 Applications of Lie (super)algebras to physics, etc.
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[1] Chen, H., Guo, X., Zhao, K.: Tensor product weight modules over the virasoro algebra. J. Lond. Math. Soci., 88(2), 829-834 (2013) · Zbl 1311.17004 · doi:10.1112/jlms/jdt046
[2] Mathieu, O.: Classification of Harish-Chandra modules over the Virasoro algebra. Invent. Math., 107, 225-234 (1992) · Zbl 0779.17025 · doi:10.1007/BF01231888
[3] Mazorchuk, V., Zhao, K.: Classification of simple weight Virasoro modules with a finite-dimensional weight space. J. Algebra, 307, 209-214 (2007) · Zbl 1128.17021 · doi:10.1016/j.jalgebra.2006.05.007
[4] Mazorchuk, V., Zhao, K.: Simple Virasoro modules which are locally finite over a positive part. Selecta Math. (N.S.), 20(3), 839-854 (2014) · Zbl 1317.17029 · doi:10.1007/s00029-013-0140-8
[5] Ondrus, M., Wiesner, E.: Whittaker modules for the Virasoro algebra. J. Algebra Appl., 8, 363-377 (2009) · Zbl 1220.17019 · doi:10.1142/S0219498809003370
[6] Tan, H., Zhao, K.: Irreducible Virasoro modules from tensor products (II). J. Algebra, 394, 357-373 (2013) · Zbl 1317.17030 · doi:10.1016/j.jalgebra.2013.07.023
[7] Zhang, H., Zhao, K.: Representations of the Virasoro-like Lie algebra and its q-analog. Comm. Algebra, 24(14), 4361-4372 (1996) · Zbl 0891.17016 · doi:10.1080/00927879608825820
[8] Osborn, J. M., Passman, D. S.: Derivations of skew polynomial rings. J. Algebra, 176, 417-448 (1995) · Zbl 0865.16020 · doi:10.1006/jabr.1995.1252
[9] Lin, W., Tan, S.: Representations of the Lie algebra of derivations for quantum torus. J. Algebra, 275, 250-274 (2004) · Zbl 1106.17020 · doi:10.1016/j.jalgebra.2003.12.021
[10] Jiang, C., Meng, D.: The automorphism group of the derivation algebra of the Virasoro-like algebra. Adv. Math. (China), 27(2), 175-183 (1998) · Zbl 1054.17505
[11] Lin, W., Tan, S.: Nonzero level Harish-Chandra modules over the Virasoro-like algebra. J. Pure Appl. Algebra, 204, 90-105 (2006) · Zbl 1105.17014 · doi:10.1016/j.jpaa.2005.03.002
[12] Kirkman, E., Procesi, C., Small, L.: A q-analog for the Virasoro algebra. Comm. Algebra, 22, 3755-3774 (1994) · Zbl 0813.17009 · doi:10.1080/00927879408825052
[13] Wang, X., Zhao, K.: Verma modules over the Virasoro-like algebra. J. Austral. Math., 80, 179-191 (2006) · Zbl 1109.17013 · doi:10.1017/S1446788700013069
[14] Arnal, D., Pinczon, G.: On algebraically irreducible representations of the Lie algebra sl2. J. Math. Phys., 15, 350-359 (1974) · Zbl 0298.17003 · doi:10.1063/1.1666651
[15] Kostant, B.: On Whittaker vectors and representation theory. Invent. Math., 48, 101-184 (1978) · Zbl 0405.22013 · doi:10.1007/BF01390249
[16] Block, R.: The irreducible representations of the Lie algebra sl2 and of the Weyl algebra. Adv. Math., 39, 69-110 (1981) · Zbl 0454.17005 · doi:10.1016/0001-8708(81)90058-X
[17] Benkart, G., Ondrus, M.: Whittaker modules for generalized Weyl algebras. Representation Theory, 13, 141-164 (2009) · Zbl 1251.16020 · doi:10.1090/S1088-4165-09-00347-1
[18] Christodoulopoulou, K.: Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine lie algebras. J. Algebra, 320, 2871-2890 (2008) · Zbl 1221.17009 · doi:10.1016/j.jalgebra.2008.06.025
[19] Wang, B., Zhu, X.: Whittaker modules for a Lie algebra of Block type. Front. Math. China, 6(4), 731-744 (2011) · Zbl 1282.17012 · doi:10.1007/s11464-011-0121-1
[20] Zhang, X., Tan, S., Lian, H.: Whittaker modules for the Schr¨oinger-Witt algebra. J. Math. Phys., 51, 083524 (2010) · Zbl 1312.81124 · doi:10.1063/1.3474916
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