Constructing Groups Associated to Infinite-Dimensional Lie Algebras

  • Victor G. Kac
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 4)


In these notes a representation theoretical approach to the construction of groups associated to (possibly infinite-dimensional) “integrable” Lie algebras is discussed. In the first part a general framework is outlined; here most of the discussion consists of definitions, examples and open problems. Deep results are available only in the case of groups associated to Kac-Moody algebras, which are discussed in the second part; it is based on joint work with Dale Peterson [18], [19], [20], [21], [22], [26]. Extension of these results to other classes of groups, like the group of biregular automorphisms of an affine space, would provide a solution to some very difficult open problems of algebraic geometry.


Exact Sequence Weyl Group Regular Function Bruhat Order Flag Variety 
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  1. [1]
    Bernstein, I.N., Gelfand, I.M. and Gelfand, S.I., Schubert cells and flag space cohomology, Uspechi Matem. Nauk 28 (1973), 3–26.MathSciNetMATHGoogle Scholar
  2. [2]
    Borel, A., Linear algebraic groups, Benjamin, New York, 1969.MATHGoogle Scholar
  3. [3]
    Bott, R., An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France 84 (1956), 251–281.MathSciNetMATHGoogle Scholar
  4. [4]
    Bourbaki, N., Groupes et Algebres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968MATHGoogle Scholar
  5. [5]
    Curtis, C.W., Central extensions of groups of Lie type, Journal für die Reine und angewandte Math., 220 (1965), 174–185.MathSciNetMATHGoogle Scholar
  6. [6]
    Deligne, P., Griffits, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds, Inventiones Math. 29 (1975), 245–274.MATHCrossRefGoogle Scholar
  7. [7]
    Garland, H., Arithmetic theory of loop groups, Publ. Math. IHES 52 (1980), 5–136.MathSciNetMATHGoogle Scholar
  8. [8]
    Garland, H. and Raghunathan, M.S., A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott, Proc. Natl. Acad. Sci. USA 72 (1975), 4716–4717.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Gohberg, I. and Feldman, I.A., Convolution equations and projection methods for their solution, Transl. Math. Monography 41, Amer. Math. Soc., Providence 1974.MATHGoogle Scholar
  10. [10]
    Grothendieck, A., Sur la classification des fibres holomorphes sur la sphere de Riemann, Amer. J. Math. 79 (1957), 121–138.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Haddad, A., A Coxeter group approach to Schubert varieties, these proceedings.Google Scholar
  12. [12]
    Kac, V.G., Simple irreducible graded lie algebras of finite growth, Math. USSR-Izvestija 2 (1968), 1271–1311.MATHCrossRefGoogle Scholar
  13. [13]
    Kac, V.G., Algebraic definition of compact Lie groups, Trudy MIEM 5 (1969), 36–47 (in Russian).Google Scholar
  14. [14]
    Kac, V.G., Infinite dimensional Lie algebras, Progess in Math. 44, Birkhauser, Boston, 1983.MATHGoogle Scholar
  15. [15]
    Kac, V.G., Torsion in cohomology of compact Lie groups and Chow rings of algebraic groups, Invent. Math., 80 (1985), 69–79.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Kac, V.G., Kazhdan, D.A., Lepowsky, J. and Wilson, R.L., Realization of the basic representation of the Euclidean Lie algebras, Advances in Math., 42 (1981), 83–112.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Kac, V.G. and Peterson, D.H., Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), 125–264.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Kac, V.G. and Peterson, D.H., Regular functions on certain infinite-dimensional groups. In: Arithmetic and Geometry, pp. 141–166. Progress in Math. 36, Birkhäuser, Boston, 1983.Google Scholar
  19. [19]
    Kac, V.G. and Peterson, D.H., Unitary structure in representations of infinite-dimensional groups and a convexity theorem, Invent. Math. 76 (1984), 1–14.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Kac, V.G. and Peterson, D.H., Defining relations of infinite-dimensional groups, Proceedings of the E. Cartan conference, Lyon, 1984.Google Scholar
  21. [21]
    Kac, V.G. and Peterson, D.H., Cohomology of infinite-dimensional groups and their flag varieties, to appear.Google Scholar
  22. [22]
    Kac, V.G., Peterson, D.H., Generalized invariants of groups generated by reflections, Proceedings of the conference. “Giornate di Geometria”, Rome, 1984.Google Scholar
  23. [23]
    Kassel, C., Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, J. Pure Applied Algebra (1984).Google Scholar
  24. [24]
    Kumar, S., Geometry of Schubert cells and cohomology of Kac-Moody Lie algebras, Journal of Diff. Geometry, (1985).Google Scholar
  25. [25]
    Moody, R., A simplicity theorem for Chevalley groups defined by generalized Cartan matrices, preprint.Google Scholar
  26. [26]
    Peterson, D.H. and Kac, V.G., Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 1778–1782.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Pressley, A., Loop groups, Grassmanians and KdV equations, these proceedings.Google Scholar
  28. [28]
    Rudakov, A.N., Automorphism groups of infinite-dimensional simple Lie algebras, Izvestija ANSSSR, (Ser. Mat.) 33 (1969), 748–764.MathSciNetGoogle Scholar
  29. [29]
    Séminair “Sophus Lie”, 1954/55. Ecole Normale Supérieure, 1955.Google Scholar
  30. [30]
    Shafarevich, I.R., On some infinite-dimensional groups II, Izvestija AN SSSR (Ser. Mat.) 45 (1981), 216–226.Google Scholar
  31. [31]
    Slodowy, P., An adjoint quotient for certain groups attached to Kac-Moody algebras, these proceedings.Google Scholar
  32. [32]
    Steinberg, R., Lectures on Chevalley groups, Yale University Lecture Notes, 1967.Google Scholar
  33. [33]
    Tits, J., Resumé de cours, College de France, Paris, 1981.Google Scholar
  34. [34]
    Tits, J., Resumé de cours, College de France, Paris, 1982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Victor G. Kac
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA

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