Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Classification of quasifinite modules over Lie algebras of matrix differential operators on the circle


Author: Yucai Su
Journal: Proc. Amer. Math. Soc. 133 (2005), 1949-1957
MSC (2000): Primary 17B10, 17B65, 17B66, 17B68
DOI: https://doi.org/10.1090/S0002-9939-05-07881-0
Published electronically: January 31, 2005
MathSciNet review: 2137860
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that an irreducible quasifinite module over the central extension of the Lie algebra of $N\times N$-matrix differential operators on the circle is either a highest or lowest weight module or else a module of the intermediate series. Furthermore, we give a complete classification of indecomposable uniformly bounded modules.


References [Enhancements On Off] (What's this?)

  • 1. B. Bakalov, V. G. Kac, A. A. Voronov, Cohomology of conformal algebras, Comm. Math. Phys. 200 (1999), 561-598. MR 1675121 (2000f:17028)
  • 2. C. Boyallian, V. Kac, J. Liberati and C. Yan, Quasifinite highest weight modules of the Lie algebra of matrix differential operators on the circle, J. Math. Phys. 39 (1998), 2910-2928. MR 1621470 (99c:17012)
  • 3. V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), 317-335. MR 0846931 (88a:17034)
  • 4. E. Frenkel, V. Kac, R. Radul and W. Wang, $\mathcal{W}_{1+\infty}$ and $\mathcal{W}(gl_N)$ with central charge $N$, Comm. Math. Phys. 170 (1995), 337-357. MR 1334399 (96i:17024)
  • 5. V. G. Kac, Vertex algebras for beginners, American Mathematical Society, Providence, 1996. MR 1651389 (99f:17033)
  • 6. V. G. Kac, The idea of locality, in ``Physical applications and mathematical aspects of geometry, groups and algebras'', H.-D. Doebner et al, eds., World Sci., Singapore, 1997, 16-32.
  • 7. V. G. Kac, Formal distribution algebras and conformal algebras, a talk at the Brisbane, in Proc. XIIth International Congress of Mathematical Physics (ICMP '97) (Brisbane), 80-97. MR 1697266 (2000f:17041)
  • 8. V. G. Kac and A. Radul, Quasi-finite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), 429-457. MR 1243706 (95f:81036)
  • 9. V. G. Kac and A. Radul, Representation theory of the vertex algebra $\mathcal{W}_{1+\infty}$, Trans. Groups 1 (1996), 41-70. MR 1390749 (97f:17033)
  • 10. V. G. Kac, W. Wang and C. H. Yan, Quasifinite representations of classical Lie subalgebras of $\mathcal{W}_{1+\infty}$, Adv. Math. 139 (1998), 46-140. MR 1652526 (2000g:17039)
  • 11. W. Li, 2-Cocycles on the algebra of differential operators, J. Alg. 122 (1989), 64-80. MR 0994935 (90d:17018)
  • 12. O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), 225-234. MR 1144422 (93d:17034)
  • 13. Y. Su, Classification of quasifinite modules over the Lie algebras of Weyl type, Adv. Math. 174 (2003), 57-68. MR 1959891 (2003m:17006)
  • 14. Y. Su, Classification of Harish-Chandra modules over the higher rank Virasoro algebras, Comm. Math. Phys. 240 (2003), 539-551.MR 2005858 (2004g:17023)
  • 15. Y. Su and K. Zhao, Isomorphism classes and automorphism groups of algebras of Weyl type, Science in China A 45 (2002), 953-963. MR 1942909 (2003j:17022)
  • 16. X. Xu, Equivalence of conformal superalgebras to Hamiltonian superoperators, Alg. Colloq. 8 (2001), 63-92. MR 1885526 (2003f:17040)
  • 17. X. Xu, Simple conformal algebras generated by Jordan algebras, preprint, math.QA/0008224.
  • 18. X. Xu, Simple conformal superalgebras of finite growth, Alg. Colloq. 7 (2000), 205-240. MR 1811245 (2002c:17043)
  • 19. X. Xu, Quadratic Conformal Superalgebras, J. Alg. 231 (2000), 1-38.MR 1779590 (2001j:17042a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17B10, 17B65, 17B66, 17B68

Retrieve articles in all journals with MSC (2000): 17B10, 17B65, 17B66, 17B68


Additional Information

Yucai Su
Affiliation: Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China — and — Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: ycsu@sjtu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-05-07881-0
Received by editor(s): February 3, 2003
Received by editor(s) in revised form: April 1, 2004
Published electronically: January 31, 2005
Additional Notes: The author was supported by an NSF grant 10171064 of China and two grants, “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents”, from the Ministry of Education of China.
Communicated by: Dan M. Barbasch
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society