Generalizations of Lie Groups

  • A. L. Onishchik
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 20)


This chapter contains a brief survey of certain theories generalizing the classical Lie theory. This survey does not aspire to completeness. In particular, we do not touch on the theory of algebraic groups and topological groups are considered only in connection with Hilbert’s 5th problem. These subjects, naturally, require separate surveys. The same can be said about “infinite” continuous groups (or Lie pseudo-groups), the study of which was originated already by S. Lie. Beyond the scope of this survey remain the Kac-Moody algebras and the groups corresponding to them, as well as Lie supergroups and superalgebras.


Moufang Loop Frechet Space Geodesic Loop Tangent Algebra Local Analytic Loop 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, M., Ratiu, T., Schmid, R. (1985): The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications. Infinite-dimensional groups with applications. Publ., Math. Sci. Res. Inst. 4, 1–69. Zbl. 617. 58004MathSciNetCrossRefGoogle Scholar
  2. Akivis, M. A. (1976): On local algebras of a multi-dimensional three-web. Sib. Mat. Zh. 17, No. 1, 5–11. Engl. transl.: Sib. Math. J. 17, No. 1, 3–8. Zbl. 337. 53018MathSciNetGoogle Scholar
  3. Akivis, M. A. (1978): On geodesic loops and local triple systems of spaces with affine connection. Sib. Mat. Zh. 19, No. 2, 243–253. Zbl. 388. 53007. Engl. transl.: Sib. Math. J. 19, No. 2, 171–178MathSciNetzbMATHGoogle Scholar
  4. Alekseevskij, D. V. (1974): Lie groups and homogeneous spaces. Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 11, 37–123. Zbl. 296. 22001. Engl. transl.: J. Sov. Math. 4, 483–539Google Scholar
  5. Alekseevskij, D. V. (1982): Lie groups. Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 20, 153–192. Zbl. 532. 22010. Engl. transl.: J. Sov. Math. 28, 924–949 (1985)Google Scholar
  6. Bourbaki, N. (1947): Topologie générale, Ch. 5–8. Hermann, Paris. Zbl. 30, 241zbMATHGoogle Scholar
  7. Bourbaki, N. (1964): Algèbre commutative, Ch. 5, 6. Hermann, Paris. Zbl. 205, 343zbMATHGoogle Scholar
  8. Bourbaki, N (1971, 1972): Groupes et algèbres de Lie. Ch. 1, 2, 3. Hermann, Paris. Zbl. 213, 41zbMATHGoogle Scholar
  9. Bruck, R. H. (1958): A Survey of Binary Systems. Springer, Berlin. Zbl. 81, 17CrossRefzbMATHGoogle Scholar
  10. Chevalley, C. (1946): Theory of Lie Groups. Vol.I. Princeton Univ. Press, PrincetonzbMATHGoogle Scholar
  11. Dieudonné, J. (1960): Foundations of Modern Analysis. Academic Press, New York, London. Zbl. 100, 42zbMATHGoogle Scholar
  12. Dieudonné, J. (1973): Introduction to the Theory of Formal Groups. Pure Appl. Math., No. 20. Marcel Dekker, New York. Zbl. 287. 14013Google Scholar
  13. Dixmier, J. (1974): Algèbres enveloppantes. Gauthier-Villars, Paris. Zbl. 308. 17007zbMATHGoogle Scholar
  14. Dynkin, E. B. (1950): Normed Lie algebras and analytic groups. Usp. Mat. Nauk. 5, No. 1, 135–186. Zbl. 41, 367MathSciNetzbMATHGoogle Scholar
  15. Dynkin, E. B. (1959): Theory of Lie Groups. Mathematics in the USSR during the Four Decades 1917–1957. Vol. 1. Fizmatgiz, Moscow, 213–227 (Russian)Google Scholar
  16. Est, W. T. van, Korthagen Th. J. (1964): Nonenlargable Lie algebras. Proc. K. Nederl. Akad. Wet., Ser. A 67, No. 1, 15–31. Zbl. 121, 275zbMATHGoogle Scholar
  17. Gelfand, I. M., Kirillov, A. A. (1966): Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. Publ. Math., Inst. Hautes Etud. Sci. 31, 5–19. Zbl. 144, 21MathSciNetCrossRefGoogle Scholar
  18. Gelfand, I. M., Kirillov, A. A. (1969): The structure of the Lie skew field connected with a semi-simple solvable Lie algebra. Funkts. Anal. Prilozh. 3, No. 1, 7–26. Engl. transl.: Funct. Anal. Appl. 3, 6–21. Zbl. 244. 17007MathSciNetGoogle Scholar
  19. Gleason, A. (1952): Groups without small subgroups. Ann. Math., II. Ser., 56, No. 2, 193–212. Zbl. 49, 301MathSciNetCrossRefzbMATHGoogle Scholar
  20. Glushkov, V. M. (1957): The structure of locally bicompact groups and Hubert’s Fifth Problem. Usp. Mat. Nauk. 12, No. 2, 3–41. Zbl. 79, 42. Engl. transl.: Transl., II. Ser., Am. Math. Soc. 15, 55–93 (1960)zbMATHGoogle Scholar
  21. Hamilton, R. S. (1982): The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc, New Ser. 7, No. 1, 65–222. Zbl. 499. 58003Google Scholar
  22. Harpe, P. de la (1972): Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space. Lect. Notes Math. 285, Springer, Berlin. Zbl. 256. 22015Google Scholar
  23. Hazewinkel, M. (1978): Formal Groups and Applications. Acad. Press, New York. Zbl. 454. 14020zbMATHGoogle Scholar
  24. Helgason, S. (1962): Differential Geometry and Symmetric Spaces. Academic Press, New York, London. Zbl. 111, 181zbMATHGoogle Scholar
  25. Humphreys, J. E. (1972): Introduction to Lie Algebras and Representation Theory. Springer, New York. Zbl. 254. 17004CrossRefzbMATHGoogle Scholar
  26. Jacobson, N. (1962): Lie Algebras. Interscience Publ. New York, London. Zbl. 121, 275zbMATHGoogle Scholar
  27. Jantzen, J. C. (1983): Einhüllende Algebren Halbeinfacher Lie Algebren. Springer, Berlin. Zbl. 541. 17001CrossRefzbMATHGoogle Scholar
  28. Joseph, A. (1974): Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras. Proc. Am. Math. Soc. 45, No. 1, 1–10. Zbl. 293. 17006CrossRefzbMATHGoogle Scholar
  29. Kaplansky, I. (1971): Lie Algebras and Locally Compact Groups. Chicago University Press, Chicago. Zbl. 223. 17001zbMATHGoogle Scholar
  30. Kirillov, A. A. (1972): Elements of Representation Theory. Nauka, Moscow. English transl.: Springer, Berlin 1976. Zbl. 342. 22001Google Scholar
  31. Kuz’min, E. N. (1968): Maltsev algebras and their representations. Algebra Logika 7, No. 4, 48–69. Zbl. 204, 361. English transl.: Algebra Logic 7, 233–244MathSciNetGoogle Scholar
  32. Kuz’min, E. N. (1971): On the relation between Maltsev algebras and analytic Moufang loops. Algebra Logika 10, No. 1, 3–22. Zbl. 244. 17019. English transl.: Algebra Logic 10, 1–14MathSciNetGoogle Scholar
  33. Lazard, M. (1952): Sur les algèbres enveloppantes universelles de certaines algèbres de Lie. C. R. Acad. Sci., Paris 234, No. 8, 788–791. Zbl. 46, 34MathSciNetzbMATHGoogle Scholar
  34. Lazard, M. (1955): Sur les groupes de Lie formels à un paramètre. Bull. Soc. Math. Fr. 83, No. 3, 251–274. Zbl. 68, 257MathSciNetzbMATHGoogle Scholar
  35. Lazard, M. (1965): Groupes analytiques p-adiques. Publ. Math., Inst. Hautes Etud. Sci. 26, 389–603. Zbl. 139, 23MathSciNetzbMATHGoogle Scholar
  36. Lazard, M., Tits, J. (1965/66): Domaines d’injectivité de l’application exponentielle. Topology 4, 315–322. Zbl. 156, 32MathSciNetCrossRefGoogle Scholar
  37. Leslie, J. (1967): On differential structure for the group of diffeomorphisms. Topology 6, 264–271MathSciNetCrossRefGoogle Scholar
  38. Leslie, J. (1992): Some integrable subalgebras of Lie algebras of infinite-dimensional Lie groups. Trans. Amer. Math. Soc. 333, No. 1, 423–443MathSciNetzbMATHGoogle Scholar
  39. Maltsev, A. I. (1945, 1946): On the theory of Lie groups in the large. Mat. Sb., Nov. Ser. 16, No. 2, 163–190, 19, No. 3, 523–524. Zbl. 161, 46Google Scholar
  40. Maltsev, A. I. (1948): Topological algebra and Lie groups. Mathematics in USSR during the 30 years 1917–1947. Gostekhizdat, Moscow-Leningrad, 134–158 (Russian). Zbl. 38, 152Google Scholar
  41. Maltsev, A. I. (1949): On a class of homogeneous spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 13, No. 1, 9–32 (Russian). Zbl. 34, 17Google Scholar
  42. Maltsev, A. I. (1955): Analytic loops. Mat. Sb., Nov. Ser. 36, No. 3, 569–576 (Russian). Zbl. 65, 7Google Scholar
  43. Manin, Yu. I. (1963): The theory of abelian formal groups over fields of finite characteristic. Usp. Mat. Nauk 18, No. 6, 3–90. Engl. transl.: Russian Math. Surv. 18, No. 6, 1–83 (1963). Zbl. 128, 156MathSciNetGoogle Scholar
  44. McConnell, J. C. (1974): Representations of solvable Lie algebras and the Gelfand-Kirillov conjecture. Proc. Lond. Math. Soc. 29, No. 3, 453–484. Zbl. 323. 17005MathSciNetCrossRefzbMATHGoogle Scholar
  45. Montgomery, D., Zippin, L. (1952): Small subgroups of finite-dimensional groups. Ann. Math., II. Ser. 56, No. 2, 213–241. Zbl. 49, 301MathSciNetCrossRefzbMATHGoogle Scholar
  46. Montgomery, D., Zippin, L. (1955): Topological Transformation Groups. Wiley, New York. Zbl. 68, 19zbMATHGoogle Scholar
  47. Morinaga, K., Nôno, T. (1950): On the logarithmic functions of matrices. I, II. J. Sci. Hiroshima Univ. Ser. A 14, No. 2, 107–114; No. 3, 171–179. Zbl. 54, 8 and Zbl. 45, 158MathSciNetGoogle Scholar
  48. Nôno, T. (1960): Sur l’application exponentielle dans les groupes de Lie. J. Sci. Hiroshima Univ. Ser. A 23, 311–324. Zbl. 94, 15MathSciNetzbMATHGoogle Scholar
  49. Omori, H. (1974): Infinite-dimensional Lie Transformation Groups. Lecture Notes Math. 427, Springer, Berlin. Zbl. 328. 58005Google Scholar
  50. Omori, H., Harpe, P. de la (1972): About interactions between Banach-Lie groups and finite-dimensional manifolds. J. Math. Kyoto Univ. 12, No. 3, 543–570. Zbl. 271. 58006MathSciNetzbMATHGoogle Scholar
  51. Pontryagin, L. S. (1984): Topological Groups. 4th edition. Nauka, Moscow. Zbl. 534. 22001. German transl.: Teubner, Leipzig 1957/1958zbMATHGoogle Scholar
  52. Postnikov, M. M. (1982): Lie Groups and Lie Algebras. Nauka, Moscow. Zbl. 597. 22001. French transl.: Leçons de géométrie. Groupes et algèbres de Lie. Éditions Mir, Moscou 1985zbMATHGoogle Scholar
  53. Sabinin, L. V., Mikheev, P. O. (1985): The Theory of Smooth Bol Loops. Publ. Univ. Druzhby Narodov (“Univ. of Friendship of Peoples”). Zbl. 584. 53001Google Scholar
  54. Sagle, A. A., Walde, R. E.(1973): Introduction to Lie Groups and Lie Algebras. Academic Press, New York, London. Zbl. 252. 22001zbMATHGoogle Scholar
  55. Serre, J-P. (1965): Lie Algebras and Lie Groups. Benjamin, New York, Amsterdam. Zbl. 132, 278zbMATHGoogle Scholar
  56. Sklyarenko, E. G. (1969): On Hilbert’s Fifth Problem. Hilbert Problems. Nauka, Moscow, 101–115 (Russian)Google Scholar
  57. Shirshov, A. I. (1953): On representation of Lie rings in associative rings. Usp. Mat. Nauk 8, No. 5, 173–175 (Russian). Zbl. 52, 30zbMATHGoogle Scholar
  58. Spanier, E. H. (1966): Algebraic Topology. McGraw Hill Book Co., New York. Zbl. 145, 433zbMATHGoogle Scholar
  59. Vinberg, E. B. (1963): Lie groups and homogeneous spaces. Itogi Nauki Tekh., Ser. Algebra, Topologiya. 1962, 5–32 (Russian). Zbl. 132, 22Google Scholar
  60. Warner, F. W. (1983): Foundations of Differentiable Manifolds and Lie Groups. Springer, New York. Zbl. 516. 58001CrossRefzbMATHGoogle Scholar
  61. Yamabe, H. (1950): On an arcwise connected subgroup of a Lie group. Osaka Math. J. 2, 13–14. Zbl. 39, 21MathSciNetzbMATHGoogle Scholar
  62. Yamabe, H. (1953a): On the conjecture of Iwasawa and Gleason. Ann. Math., II. Ser. 58, No. 1, 48–54. Zbl. 53, 16MathSciNetCrossRefzbMATHGoogle Scholar
  63. Yamabe, H. (1953b): A generalization of a theorem of Gleason. Ann. Math., II. Ser. 58, No. 2, 351–365. Zbl. 53, 16MathSciNetCrossRefzbMATHGoogle Scholar
  64. Yang, C. T. (1976): Hilbert’s fifth problem and related problems on transformation groups. Mathematical Developements Arising from Hilbert Problems. Proc. Symp. Pure Math. 28, Am. Math. Soc., Providence 142–146. Zbl. 362. 57006Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • A. L. Onishchik
    • 1
  1. 1.Yaroslavl UniversityYaroslavlRussia

Personalised recommendations