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)



Double Koszul complex and construction of irreducible representations of $ \mathfrak{gl}(3\vert 1)$

Author: Nguyên Thi Phuong Dung
Journal: Proc. Amer. Math. Soc. 138 (2010), 3783-3796
MSC (2000): Primary 17B10, 17B70; Secondary 20G05, 20G42
Published electronically: May 24, 2010
MathSciNet review: 2679601
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ V$ be a super vector space with super dimension $ (m\vert n)$. Manin introduced the Koszul complex associated to $ V$, which is denoted $ K$. There is another Koszul complex, denoted $ L$. Our observation is that these two Koszul complexes can be combined into a double complex, which we call the double Koszul complex. By using the differential of this complex, we give a way to describe all irreducible representations of $ \frak {gl}(V)$ when $ V$ has super dimension $ (3\vert 1)$.

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

  • 1. A. Berele and A. Regev.
    Hook Young Diagrams with Applications to Combinatorics and to Representations of Lie Superalgebras.
    Advances in Math., 64:118-175, 1987. MR 884183 (88i:20006)
  • 2. J. Brundan.
    Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $ gl(m\vert n)$.
    J. Amer. Math. Soc., 16:185-231, 2002. MR 1937204 (2003k:17007)
  • 3. N.T.P. Dung, P.H. Hai.
    Irreducible Representations of Quantum Linear Groups of Type $ A_{1\vert}$.
    J. Alg., 282, 809 - 830, 2004. MR 2101085 (2006c:16074)
  • 4. D.I. Gurevich.
    Algebraic aspects of the quantum Yang-Baxter equation.
    Leningrad Math. J., 2(4)801-828, 1987. MR 1080202 (93e:17018)
  • 5. P.H. Hai.
    Splitting comodules over Hopf algebras and application to representation theory of quantum groups of type $ A\sb {0\vert 0}$.
    J. of Algebra, 245(1):20-41, 2001. MR 1868181 (2002j:16045)
  • 6. V.G. Kac.
    Classification of simple Lie superalgebras.
    Funct. Anal. Appl., 9:263-265, 1975. MR 0390005 (52:10833)
  • 7. V.G. Kac.
    Lie superalgebras.
    Adv. Math., 26:8-96, 1977. MR 0486011 (58:5803)
  • 8. V.G. Kac.
    Character of typical representations of classical Lie superalgebras.
    Comm. Alg., 5:889-897, 1977. MR 0444725 (56:3075)
  • 9. V.G. Kac.
    Representations of classical Lie superalgebras,
    in: Lecture Notes in Math., 676:597-626, Springer, 1978. MR 519631 (80f:17006)
  • 10. I.G. Macdonald.
    Symmetric Functions and Hall Polynomials.
    Oxford University Press, New York, 1979. MR 553598 (84g:05003)
  • 11. Yu.I. Manin.
    Quantum Groups and Noncommutative Geometry.
    Université de Montréal Centre de Recherches Mathématiques, Quebec, 1988. MR 1016381 (91e:17001)
  • 12. Yu.I. Manin.
    Gauss Field Theory and Complex Geometry.
    Springer-Verlag, 1988. MR 954833 (89d:32001)
  • 13. M. Scheunert.
    The Theory of Lie Superalgebras.
    Lecture Notes in Math., Springer-Verlag, 1978. MR 537441 (80i:17005)
  • 14. M. Scheunert, R.B. Zhang.
    The general linear supergroup and its Hopf superalgebra of regular functions.
    Jour. Alg., 254:44-83, 2002. MR 1927431 (2003h:16069)
  • 15. Yucai Su, R.B. Zhang.
    Character and dimension formulae for general linear superalgebra.
    Adv. Math., 211:1-33, 2007. MR 2313526 (2008f:17012)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17B10, 17B70, 20G05, 20G42

Retrieve articles in all journals with MSC (2000): 17B10, 17B70, 20G05, 20G42

Additional Information

Nguyên Thi Phuong Dung
Affiliation: Department of Algebra, Institute of Mathematics, VAST, 18 Hoang Quôc Viet Road, CauGiay, 10307, Ha Noi, Viet Nam

Received by editor(s): October 15, 2009
Received by editor(s) in revised form: January 21, 2010
Published electronically: May 24, 2010
Additional Notes: Financial support provided to the author by NAFOSTED under grant no.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society