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Transformation Groups

, Volume 20, Issue 1, pp 183–228 | Cite as

EXTENSIONS AND BLOCK DECOMPOSITIONS FOR FINITE-DIMENSIONAL REPRESENTATIONS OF EQUIVARIANT MAP ALGEBRAS

  • ERHARD NEHER
  • ALISTAIR SAVAGE
Article

Abstract

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. The irreducible finite-dimensional representations of these algebras were classified in [NSS12], where it was shown that they are all tensor products of evaluation representations and one-dimensional representations.

In the current paper, we describe the extensions between irreducible finite-dimensional representations of an equivariant map algebra in the case that X is an affine scheme of finite type and g is reductive. This allows us to also describe explicitly the blocks of the category of finite-dimensional representations in terms of spectral characters, whose definition we extend to this general setting. Applying our results to the case of generalized current algebras (the case where the group acting is trivial), we recover known results but with very different proofs. For (twisted) loop algebras, we recover known results on block decompositions (again with very different proofs) and new explicit formulas for extensions. Finally, specializing our results to the case of (twisted) multiloop algebras and generalized Onsager algebras yields previously unknown results on both extensions and block decompositions.

Keywords

Isomorphism Class Evaluation Representation Spectral Character Block Decomposition Invariant Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AM69]
    M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969. Russian transl.: M. Атья, И. Макдональд, Введение в коммуmаmиую алгебру, Мир, М., 1972.Google Scholar
  2. [Bou61]
    N. Bourbaki, Éléments de Mathématique. Fasc. XXVII. Algèbre Commutative, Chap. 1: Modules Plats, Chap. 2: Localisation, Actualités Scientifiques et Industrielles, No. 1290, Herman, Paris, 1961. Russian transl.: Н. Бурбаки, Коммуmаmивная алгебра, Мир, M., 1971.Google Scholar
  3. [Bou71]
    N. Bourbaki. Éléments de Mathématique. Fasc. XXVI. Groupes et Algèbres de Lie, Chap. I: Algèbres de Lie, Seconde éd., Actualités Scientifiques et Industrielles, No. 1285, Paris, Hermann, 1971. Russian transl.: Н. Бурбаки, Груnnыи алгебры Ли, Мир, M., 1976.Google Scholar
  4. [Bou75]
    N. Bourbaki. Éléments de Mathématique. Fasc. XXXVIII, Groupes et Algèbres de Lie, Chap. VII: Sous-Algèbres de Cartan, Éements Réguliers, Chap. VIII: Algèbres de Lie Semi-Simples Déployées, Actualités Scientifiques et Industrielles, No. 1364. Paris, Hermann, 1975. Russian transl.: Н. Бурбаки, Груnnыи алгебры Ли, Мир, M., 1978.Google Scholar
  5. [Bou81]
    N. Bourbaki. Éléments de Mathématique. Groupes et Algèbres de Lie, Chap. 4, 5 et 6, Paris, Masson, 1981.Google Scholar
  6. [Bou85]
    N. Bourbaki. Éléments de Mathématique. Algèbre Commutative, Chap. 5 à 7, reprint, Paris, Masson, 1985.Google Scholar
  7. [CFK10]
    V. Chari, G. Fourier, T. Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517–549.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [CG05]
    V. Chari, J. Greenstein, An application of free Lie algebras to polynomial current algebras and their representation theory, in: Infinitedimensional Aspects of Representation Theory and Applications, Contemp. Math., Vol. 392, Amer. Math. Soc., Providence, RI, 2005, pp. 15–31.Google Scholar
  9. [CM04]
    V. Chari, A. A. Moura, Spectral characters of finite-dimensional representations of affine algebras, J. Algebra 279 (2004), no. 2, 820–839.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [FM94]
    A. Fialowski, F. Malikov, Extensions of modules over loop algebras, Amer. J. Math. 116 (1994), no. 5, 1265–1281.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [Fuk86]
    Д. Б. фукс, Когомолгии бесконмерных алгебр Ли, Наука, M., 1984. Engl. transl.: D. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.Google Scholar
  12. [GW09]
    R. Goodman, N. R. Wallach, Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, Vol. 255, Springer, Dordrecht, 2009.Google Scholar
  13. [Hel01]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, corrected reprint of the 1978 original, American Mathematical Society, Providence, RI, 2001.Google Scholar
  14. [HS53]
    G. Hochschild, J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603.Google Scholar
  15. [Hum72]
    J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York, 1972. Russian transl.: Дж. Хамфрис, Введение в mеорию алгебр Ли, и, их nредсmавлений, МЦНМО, M., 2003.Google Scholar
  16. [Jan03]
    J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
  17. [Kac90]
    V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed, Cambridge University Press, Cambridge, 1990. Russian transl.: В. Кац, Бесконечномерные, алгебры Ли, Мир, M., 1993.Google Scholar
  18. [Kod10]
    R. Kodera, Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Transform. Groups 15 (2010), no. 2, 371–388.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [Kum02]
    S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.Google Scholar
  20. [Lau10]
    M. Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1, 167–184.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [NSS12]
    E. Neher, A. Savage, P. Senesi, Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2619-2646.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [Roa91]
    S.-s. Roan, Onsagers algebra, loop algebra and chiral Pots model, Max- Planck-Institut preprint MPI/91-70, 1991.Google Scholar
  23. [Sol]
    Séminaire Sophus Lie de l’Ecole Normale Supérieure, 1954/1955. Théorie des Algèbres de Lie. Topologie des Groupes de Lie Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1955. Russian transl.: Теория алгебр Ли. Тоnология груnn Ли. СеминарСофус Ли”, ИЛ, M., 1962.Google Scholar
  24. [Sen10]
    P. Senesi, The block decomposition of finite-dimensional representations of twisted loop algebras, Pacific J. Math. 244 (2010), no. 2, 335–357.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [Ste68]
    R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, 1968. Russian transl.: Р. Стейнберг, Лекции о груnnах Шевалле, Мир, M., 1975.Google Scholar
  26. [Wei94]
    C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of OttawaOttawaCanada

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