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Mathematische Zeitschrift

, Volume 277, Issue 1–2, pp 373–399 | Cite as

Equivariant map superalgebras

  • Alistair Savage
Article

Abstract

Suppose a group \(\Gamma \) acts on a scheme \(X\) and a Lie superalgebra \(\mathfrak {g}\). The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from \(X\) to \(\mathfrak {g}\). We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of \(X\) is finitely generated, \(\Gamma \) is finite abelian and acts freely on the rational points of \(X\), and \(\mathfrak {g}\) is a basic classical Lie superalgebra (or \(\mathfrak {sl}\,(n,n)\), \(n \ge 1\), if \(\Gamma \) is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on \(X\). Furthermore, in the case that the even part of \(\mathfrak {g}\) is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of \(\mathfrak {g}\) is not semisimple (more generally, if \(\mathfrak {g}\) is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.

Keywords

Lie superalgebra Basic classical Lie superalgebra Loop superalgebra Equivariant map superalgebra Finite dimensional representation  Finite dimensional module 

Mathematics Subject Classification (2010)

17B65 17B10 

Notes

Acknowledgments

The author would like to thank Shun-Jen Cheng, Daniel Daigle, Dimitar Grantcharov, Erhard Neher and Hadi Salmasian for helpful discussions.

References

  1. 1.
    Atiyah, M. F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading, London-Don Mills, Ont. (1969)Google Scholar
  2. 2.
    Batra, P.: Representations of twisted multi-loop Lie algebras. J. Algebr. 272(1), 404–416 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Chari, V., Fourier, G., Khandai, T.: A categorical approach to Weyl modules. Transform. Groups 15(3), 517–549 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chari, V., Fourier, G., Senesi, P.: Weyl modules for the twisted loop algebras. J. Algebr. 319(12), 5016–5038 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Chari, V.: Integrable representations of affine Lie-algebras. Invent. Math. 85(2), 317–335 (1986)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cheng, S.-J.: Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras. J. Algebr. 173(1), 1–43 (1995)CrossRefMATHGoogle Scholar
  7. 7.
    Chari, V., Moura, A.: Spectral characters of finite-dimensional representations of affine algebras. J. Algebr. 279(2), 820–839 (2004)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chari, V., Pressley, A.: New unitary representations of loop groups. Math. Ann. 275(1), 87–104 (1986)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chari, V., Pressley, A.: Twisted quantum affine algebras. Comm. Math. Phys. 196(2), 461–476 (1998)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Eisenbud, D., Harris, J.: The Geometry of Schemes, Volume 197 of Graduate Texts in Mathematics. Springer, New York (2000)Google Scholar
  11. 11.
    Eswara Rao, S.: Finite dimensional modules for multiloop superalgebra of type \({A}(m, n)\) and \({C}(m)\). arXiv:1109.3297v1 [math.RT]Google Scholar
  12. 12.
    Eswara Rao, S.: On representations of loop algebras. Comm. Algebr. 21(6), 2131–2153 (1993)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Eswara Rao, S.: Classification of irreducible integrable modules for multi-loop algebras with finite-dimensional weight spaces. J. Algebr. 246(1), 215–225 (2001)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Eswara Rao, S., Zhao, K.: On integrable representations for toroidal Lie superalgebras. In Kac-Moody Lie Algebras and Related Topics, Volume 343 of Contemp. Math., pp. 243–261. Amer. Math. Soc., Providence (2004)Google Scholar
  15. 15.
    Fourier, G., Khandai, T., Kuz, D., Savage, A.: Local weyl modules for equivariant map algebras with free abelian group actions. J. Algebr. 350, 386–404 (2012)CrossRefMATHGoogle Scholar
  16. 16.
    Feigin, B., Loktev, S.: Multi-dimensional Weyl modules and symmetric functions. Comm. Math. Phys. 251(3), 427–445 (2004)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Frappat, L., Sciarrino, A., Sorba, P.: Dictionary on Lie Algebras and Superalgebras. Academic Press, San Diego (2000)MATHGoogle Scholar
  18. 18.
    Fu, L.: Etale Cohomology Theory,Volume 13 of Nankai Tracts in Mathematics. World Scientific, Hackensack (2011)Google Scholar
  19. 19.
    Józefiak, T.: Semisimple superalgebras. In Algebra—Some Current Trends (Varna, 1986), Volume 1352 of Lecture Notes in Math., pp. 96–113. Springer, Berlin (1988)Google Scholar
  20. 20.
    Kac, V.G.: Characters of typical representations of classical Lie superalgebras. Comm. Algebr. 5(8), 889–897 (1977)CrossRefMATHGoogle Scholar
  21. 21.
    Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)CrossRefMATHGoogle Scholar
  22. 22.
    Kac, V.G.: A sketch of Lie superalgebra theory. Comm. Math. Phys. 53(1), 31–64 (1977)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Kac, V.: Representations of classical Lie superalgebras. In Differential Geometrical Methods in Mathematical Physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Volume 676 of Lecture Notes in Math., pp. 597–626. Springer, Berlin (1978)Google Scholar
  24. 24.
    Lang, S.: Algebra, Volume 211 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2002)Google Scholar
  25. 25.
    Lau, M.: Representations of multiloop algebras. Pac. J. Math. 245(1), 167–184 (2010)CrossRefMATHGoogle Scholar
  26. 26.
    Li, H.: On certain categories of modules for affine Lie algebras. Math. Z. 248(3), 635–664 (2004)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Neher, E., Savage, A.: Extensions and block decompositions for finite-dimensional representations of equivariant map algebras. arXiv:1103.4367v1 [math.RT]Google Scholar
  28. 28.
    Neher, E., Savage, A., Senesi, P.: Irreducible finite-dimensional representations of equivariant map algebras. Trans. Am. Math. Soc. 364(5), 2619–2646 (2012)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Savage, A.: Classification of irreducible quasifinite modules over map Virasoro algebras. Transform. Groups 17(2), 547–570 (2012)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction, Volume 11 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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