Abstract
We show that the support of an irreducible weight module over the twisted Heisenberg– Virasoro algebra, which has an infinite–dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg–Virasoro algebra, having a nontrivial finite–dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series).
Similar content being viewed by others
References
Arbarello, E., De Concini, C., Kac, V. G., Procesi, C.: Moduli spaces of curves and representation theory. Comm. Math. Phys., 117, 1–36 (1988)
Billig, Y.: Respresentations of the twisted Heisenberg–Virasoro algebra at level zero. Canad. Math. Bulletin, 46, 529–537 (2003)
Fabbri, M. A., Okoh, F.: Representations of Virasoro–Heisenberg algebras and Virasoro–toroidal algebras. Canad. J. Math., 51(3), 523–545 (1999)
Jiang, Q., Jiang, C.: Representations of the twisted Heisenberg–Virasoro algebra and the full toroidal Lie algebras. Algebra Colloq., accepted
Shen, R., Jiang, C.: Derivation algebra and automorphism group of the twisted Heisenberg–Virasoro algebra. Comm. Algebra, accepted
Lu, R., Zhao, K.: Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra, preprint (arXiv:math.ST/ 0510194)
Lin, S., Xin, B.: Representations of a noncommutative associative algebra related to quantum torus of rank three. Acta Mathematica Sinica, English Series, 21, 1521–1524 (2005)
Wu, Y., Su, Y.: Lie bialgebras of generalized Virasoro–like type. Acta Mathematica Sinica, English Series, 22(6), (2006)
Zhu, L., Meng, D.: Structure of degenerate Block algebras. Algebra Colloquium, 10, 53–62 (2003)
Ding, W., Wang, H., Wang, Y.: Schrodinger flows on compact Hermitian symmetric spaces and related problems. Acta Mathematica Sinica, English Series, 19(2), 303–312 (2003)
Cohn, W. S., Lu, G.: Best constants for Moser–Trudinger inequalities, fundamental solutions and oneparameter. Acta Mathematica Sinica, English Series, 18(2), 375–390 (2002)
Zhao, K.: Automorphisms of the algebra of q–difference operators. Acta Mathematica Sinica, English Series, 15(2), 145–152 (1999)
Lin, S., Wu, Y.: Representations of the quantized Weyl algebra associated to the quantum plane. Algebra Colloquium, 124, 715–720 (2005)
Dokovic, D., Zhao, K.: Derivations, isomorphisms and second cohomology of generalized Block algebras. Algebra Colloquium, 3, 245–272 (1996)
Xu, X.: Pointed representations of Virasoro algebra, A Chinese summary appears in Acta Math. Sinica, Chinese Series, 40(3), 479; Acta Math. Sinica, New Series, 13, 161–168 (1997)
Mazorchuk, V.: On simple mixed modules over the Virasoro algebra, Mat. Stud. 22, 121–128 (2004)
Mazorchuk, V., Zhao, K.: Classification of simple weight Virasoro modules with a finite–dimensional weight space, preprint (arXiv:math.RT/ 0507195)
Author information
Authors and Affiliations
Corresponding author
Additional information
*Supported by NSF Grants 10471096, 10571120 of China and “One Hundred Talents Program” from the University of Science and Technology of China
Rights and permissions
About this article
Cite this article
Shen, R., Su*, Y.C. Classification of Irreducible Weight Modules with a Finite–dimensional Weight Space over Twisted Heisenberg–Virasoro Algebra. Acta Math Sinica 23, 189–192 (2007). https://doi.org/10.1007/s10114-005-0807-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-005-0807-y