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Generalizations of Lie Groups

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Abstract

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.

Keywords

  • 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.

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References

  • 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. 58004

    MathSciNet  CrossRef  Google Scholar 

  • 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. 53018

    MathSciNet  Google Scholar 

  • 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, 171–178

    MathSciNet  MATH  Google Scholar 

  • 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, 483–539

    Google Scholar 

  • 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 

  • Bourbaki, N. (1947): Topologie générale, Ch. 5–8. Hermann, Paris. Zbl. 30, 241

    MATH  Google Scholar 

  • Bourbaki, N. (1964): Algèbre commutative, Ch. 5, 6. Hermann, Paris. Zbl. 205, 343

    MATH  Google Scholar 

  • Bourbaki, N (1971, 1972): Groupes et algèbres de Lie. Ch. 1, 2, 3. Hermann, Paris. Zbl. 213, 41

    MATH  Google Scholar 

  • Bruck, R. H. (1958): A Survey of Binary Systems. Springer, Berlin. Zbl. 81, 17

    CrossRef  MATH  Google Scholar 

  • Chevalley, C. (1946): Theory of Lie Groups. Vol.I. Princeton Univ. Press, Princeton

    MATH  Google Scholar 

  • Dieudonné, J. (1960): Foundations of Modern Analysis. Academic Press, New York, London. Zbl. 100, 42

    MATH  Google Scholar 

  • Dieudonné, J. (1973): Introduction to the Theory of Formal Groups. Pure Appl. Math., No. 20. Marcel Dekker, New York. Zbl. 287. 14013

    Google Scholar 

  • Dixmier, J. (1974): Algèbres enveloppantes. Gauthier-Villars, Paris. Zbl. 308. 17007

    MATH  Google Scholar 

  • Dynkin, E. B. (1950): Normed Lie algebras and analytic groups. Usp. Mat. Nauk. 5, No. 1, 135–186. Zbl. 41, 367

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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, 275

    MATH  Google Scholar 

  • 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, 21

    MathSciNet  CrossRef  Google Scholar 

  • 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. 17007

    MathSciNet  Google Scholar 

  • Gleason, A. (1952): Groups without small subgroups. Ann. Math., II. Ser., 56, No. 2, 193–212. Zbl. 49, 301

    MathSciNet  CrossRef  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • 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. 58003

    Google Scholar 

  • 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. 22015

    Google Scholar 

  • Hazewinkel, M. (1978): Formal Groups and Applications. Acad. Press, New York. Zbl. 454. 14020

    MATH  Google Scholar 

  • Helgason, S. (1962): Differential Geometry and Symmetric Spaces. Academic Press, New York, London. Zbl. 111, 181

    MATH  Google Scholar 

  • Humphreys, J. E. (1972): Introduction to Lie Algebras and Representation Theory. Springer, New York. Zbl. 254. 17004

    CrossRef  MATH  Google Scholar 

  • Jacobson, N. (1962): Lie Algebras. Interscience Publ. New York, London. Zbl. 121, 275

    MATH  Google Scholar 

  • Jantzen, J. C. (1983): Einhüllende Algebren Halbeinfacher Lie Algebren. Springer, Berlin. Zbl. 541. 17001

    CrossRef  MATH  Google Scholar 

  • Joseph, A. (1974): Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras. Proc. Am. Math. Soc. 45, No. 1, 1–10. Zbl. 293. 17006

    CrossRef  MATH  Google Scholar 

  • Kaplansky, I. (1971): Lie Algebras and Locally Compact Groups. Chicago University Press, Chicago. Zbl. 223. 17001

    MATH  Google Scholar 

  • Kirillov, A. A. (1972): Elements of Representation Theory. Nauka, Moscow. English transl.: Springer, Berlin 1976. Zbl. 342. 22001

    Google Scholar 

  • 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, 233–244

    MathSciNet  Google Scholar 

  • 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, 1–14

    MathSciNet  Google Scholar 

  • 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, 34

    MathSciNet  MATH  Google Scholar 

  • Lazard, M. (1955): Sur les groupes de Lie formels à un paramètre. Bull. Soc. Math. Fr. 83, No. 3, 251–274. Zbl. 68, 257

    MathSciNet  MATH  Google Scholar 

  • Lazard, M. (1965): Groupes analytiques p-adiques. Publ. Math., Inst. Hautes Etud. Sci. 26, 389–603. Zbl. 139, 23

    MathSciNet  MATH  Google Scholar 

  • Lazard, M., Tits, J. (1965/66): Domaines d’injectivité de l’application exponentielle. Topology 4, 315–322. Zbl. 156, 32

    MathSciNet  CrossRef  Google Scholar 

  • Leslie, J. (1967): On differential structure for the group of diffeomorphisms. Topology 6, 264–271

    MathSciNet  CrossRef  Google Scholar 

  • Leslie, J. (1992): Some integrable subalgebras of Lie algebras of infinite-dimensional Lie groups. Trans. Amer. Math. Soc. 333, No. 1, 423–443

    MathSciNet  MATH  Google Scholar 

  • 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, 46

    Google Scholar 

  • 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, 152

    Google Scholar 

  • Maltsev, A. I. (1949): On a class of homogeneous spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 13, No. 1, 9–32 (Russian). Zbl. 34, 17

    Google Scholar 

  • Maltsev, A. I. (1955): Analytic loops. Mat. Sb., Nov. Ser. 36, No. 3, 569–576 (Russian). Zbl. 65, 7

    Google Scholar 

  • 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, 156

    MathSciNet  Google Scholar 

  • 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. 17005

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Montgomery, D., Zippin, L. (1952): Small subgroups of finite-dimensional groups. Ann. Math., II. Ser. 56, No. 2, 213–241. Zbl. 49, 301

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Montgomery, D., Zippin, L. (1955): Topological Transformation Groups. Wiley, New York. Zbl. 68, 19

    MATH  Google Scholar 

  • 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, 158

    MathSciNet  Google Scholar 

  • Nôno, T. (1960): Sur l’application exponentielle dans les groupes de Lie. J. Sci. Hiroshima Univ. Ser. A 23, 311–324. Zbl. 94, 15

    MathSciNet  MATH  Google Scholar 

  • Omori, H. (1974): Infinite-dimensional Lie Transformation Groups. Lecture Notes Math. 427, Springer, Berlin. Zbl. 328. 58005

    Google Scholar 

  • 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. 58006

    MathSciNet  MATH  Google Scholar 

  • Pontryagin, L. S. (1984): Topological Groups. 4th edition. Nauka, Moscow. Zbl. 534. 22001. German transl.: Teubner, Leipzig 1957/1958

    MATH  Google Scholar 

  • 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 1985

    MATH  Google Scholar 

  • Sabinin, L. V., Mikheev, P. O. (1985): The Theory of Smooth Bol Loops. Publ. Univ. Druzhby Narodov (“Univ. of Friendship of Peoples”). Zbl. 584. 53001

    Google Scholar 

  • Sagle, A. A., Walde, R. E.(1973): Introduction to Lie Groups and Lie Algebras. Academic Press, New York, London. Zbl. 252. 22001

    MATH  Google Scholar 

  • Serre, J-P. (1965): Lie Algebras and Lie Groups. Benjamin, New York, Amsterdam. Zbl. 132, 278

    MATH  Google Scholar 

  • Sklyarenko, E. G. (1969): On Hilbert’s Fifth Problem. Hilbert Problems. Nauka, Moscow, 101–115 (Russian)

    Google Scholar 

  • Shirshov, A. I. (1953): On representation of Lie rings in associative rings. Usp. Mat. Nauk 8, No. 5, 173–175 (Russian). Zbl. 52, 30

    MATH  Google Scholar 

  • Spanier, E. H. (1966): Algebraic Topology. McGraw Hill Book Co., New York. Zbl. 145, 433

    MATH  Google Scholar 

  • Vinberg, E. B. (1963): Lie groups and homogeneous spaces. Itogi Nauki Tekh., Ser. Algebra, Topologiya. 1962, 5–32 (Russian). Zbl. 132, 22

    Google Scholar 

  • Warner, F. W. (1983): Foundations of Differentiable Manifolds and Lie Groups. Springer, New York. Zbl. 516. 58001

    CrossRef  MATH  Google Scholar 

  • Yamabe, H. (1950): On an arcwise connected subgroup of a Lie group. Osaka Math. J. 2, 13–14. Zbl. 39, 21

    MathSciNet  MATH  Google Scholar 

  • Yamabe, H. (1953a): On the conjecture of Iwasawa and Gleason. Ann. Math., II. Ser. 58, No. 1, 48–54. Zbl. 53, 16

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Yamabe, H. (1953b): A generalization of a theorem of Gleason. Ann. Math., II. Ser. 58, No. 2, 351–365. Zbl. 53, 16

    MathSciNet  CrossRef  MATH  Google Scholar 

  • 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. 57006

    Google Scholar 

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Onishchik, A.L. (1993). Generalizations of Lie Groups. In: Onishchik, A.L. (eds) Lie Groups and Lie Algebras I. Encyclopaedia of Mathematical Sciences, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57999-8_5

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  • DOI: https://doi.org/10.1007/978-3-642-57999-8_5

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