RESEARCH ARTICLE

Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

  • Xiufu ZHANG
Expand
  • School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received date: 08 May 2014

Accepted date: 09 Mar 2015

Published date: 24 Jun 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

There are two extensions of Virasoro algebra with particular importance in superstring theory: the Ramond algebra and the Neveu-Schwarz algebra, which are Z2-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.

Cite this article

Xiufu ZHANG . Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1233 -1242 . DOI: 10.1007/s11464-015-0466-y

1
Astashkevich A. On the structure of Verm modules over Virasoro algebra and Neveu-Schwarz algebras. Comm Math Phys, 1997, 186: 531-562

DOI

2
Chen H, Guo X, Zhao K. Tensor Product weight modules over the Virasoro algebra. J Lond Math Soc (2), 2013, 88: 829-834

3
Feigin B, Fuchs D. Representations of the Virasoro algebra. In: Representations of Lie groups and related topics. Adv Stud Contemp Math, Vol 7. New York: Gordon and Breach, 1990, 465-554

4
Guo X, Lu R, Zhao K. Irreducible modules over the Virasoro algebra. Doc Math, 2011, 16: 709-721

5
Guo X, Lu R, Zhao K. Fraction representations and highest-weight-like representations of the Virasoro algebra. J Algebra, 2013, 387: 68-86

DOI

6
Li J, Su Y. Irreducible weight modules over the twisted Schrödinger-Virasoro algebra. Acta Math Sin (Engl Ser), 2009, 25(4): 531-536

DOI

7
Liu G, Lu R, Zhao K. A class of simple weight Virasoro modules. arXiv: 1211.0998

8
Lu R, Zhao K. Irreducible Virasoro modules from irreducible Weyl modules. J Algebra, 2014, 414: 271-287

DOI

9
Lu R, Zhao K. A family of simple weight modules over the Virasoro algebra. arXiv: 1303.0702

10
Martin C, Piard A. Indecomposable modules over the Virasoro Lie algebra and a conjecture of V. Kac. Comm Math Phys, 1991, 137(1): 109-132

DOI

11
Mathieu O. Classification of Harish-Chandra modules over the Virasoro Lie algebra. Invent Math, 1992, 107: 225-234

DOI

12
Mazorchuk V, Zhao K. Classification of simple weight Virasoro modules with a finitedimensional weight space. J Algebra, 2007, 307: 209-214

DOI

13
Mazorchuk V, Zhao K. Simple Virasoro modules which are locally finite over a positive part. Selecta Math (N S), 2014, 20(3): 839-854

DOI

14
Shen R, Su Y. Classification of irreducible weight modules with a finite-dimensional weight space over twisted Heisenberg-Virasoro algebra. Acta Math Sin (Engl Ser), 2007, 23(1): 189-192

DOI

15
Su Y. A classification of indecomposable sl2(ℂ)-modules and a conjecture of Kac on irreducible modules over the Virasoro algebra. J Algebra, 1993, 161: 33-46

16
Su Y. Classification of Harish-Chandra modules over the super-Virasoro algebras. Comm Algebra, 1995, 23: 3653-3675

DOI

17
Tan H, Zhao K. Irreducible Virasoro modules from tensor products (II). J Algebra, 2013, 394: 357-373

DOI

18
Zhang H. A class of representations over the Virasoro algebra. J Algebra, 1997, 190: 1-10

DOI

19
Zhang X. Tensor product weight representations of the Neveu-Schwarz algebra. Comm Algebra (to appear)

20
Zhang X, Xia Z. Classification of simple weight modules for the Neveu-Schwarz algebra with a finite-dimensional weight space. Comm Algebra, 2012, 40: 2161-2170

DOI

21
Zhao K. Representations of the Virasoro algebra I. J Algebra, 1995, 176: 882-907

DOI

Outlines

/