Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations


Author: A. I. Shtern
Translated by: G. G. Gould
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2011, 79-95
MSC (2010): Primary 22E15, 22C05
DOI: https://doi.org/10.1090/S0077-1554-2012-00190-3
Published electronically: January 12, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a number of consequences of the theorem on the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the derived subgroup of the group, as well as an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of connected soluble locally compact groups. In particular, we give a description of connected Lie groups admitting a (not necessarily continuous) faithful locally bounded finite-dimensional representation; as it turns out, such groups are linear. Furthermore, we give a description of the intersection of the kernels of continuous finite-dimensional representations of a given connected locally compact group, obtain a generalization of Hochschild's theorem on the kernel of the universal representation in terms of locally bounded (not necessarily continuous) finite-dimensional linear representations, and find the intersection of the kernels of such representations for a connected reductive Lie group.


References [Enhancements On Off] (What's this?)

  • 1. É. Cartan, Sur les représentations linéaires des groupes clos, Comment. Math. Helv. 2 (1930), no. 1, 269-283. MR 1509418
  • 2. B.L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933), 780-786. MR 1545369
  • 3. J.D. Jawson, Intrinsic topologies in topological lattices and semilattices, Pacific J. Math. 44 (1973), 593-602. MR 0318031 (47:6580)
  • 4. W.W. Comfort, Topological groups, Handbook of set-theoretic topology, North-Holland, Amsterdam-New York, 1984, pp. 1143-1263. MR 776643 (86g:22001)
  • 5. A. Pilay, An application of model theory to real and $ p$-adic algebraic groups, J. Algebra 126 (1989), no. 1, 139-146. MR 1023289 (90m:03061)
  • 6. K.H. Hofmann and S.A. Morris, The structure of compact groups, de Gruyter, Berlin, 1998. MR 1646190 (99k:22001)
  • 7. W.W. Comfort, D. Remus and H. Szambien, Extending ring topologies, J. Algebra 232 (2000), no. 1, 21-47. MR 1783911 (2001i:16081)
  • 8. J.E. Hart and K. Kunen, Bohr compactifications of non-abelian groups, Topology Proc. 26 (2001/02), no. 2, 593-626. MR 2032839 (2005c:22012)
  • 9. A.I. Shtern, A version of van der Waerden's theorem and a proof of Mishchenko's conjecture for homomorphisms of locally compact groups, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 1, 183-224; English transl., Izv. Math. 72 (2008), no. 1, 169-205. MR 2394977 (2009h:22004)
  • 10. D. Remus, Private communication (2010).
  • 11. W.W. Comfort and L.C. Robertson, Images and quotients of $ \mathrm {SO(3,\mathbf {R})}$: remarks on a theorem of van der Waerden, Rocky Mountain J. Math. 17 (1987), no. 1, 1-13. MR 882879 (88f:22012)
  • 12. A. Borel, Essays in the history of Lie groups and algebraic groups, Amer. Math. Soc., Providence, RI: London Mathematical Society, Cambridge, 2001. MR 1847105 (2002g:01010)
  • 13. O. Schreier and B.L. van der Waerden, Die Automorphismen der projektiven Gruppen, Abh. Math. Sem. Hamburg 6 (1928), 303-322.
  • 14. H. Freudenthal, Die Topologie der Lieschen Gruppen als algebraisches Phänomenon I , Ann. of Math. (2) 42 (1941), 1051-1074. MR 0005740 (3:198a)
  • 15. W.T. van Est, Dense imbeddings of Lie groups, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 321-328. MR 0044530 (13:432c)
  • 16. W.T. van Est, Dense imbeddings of Lie groups II (I,II), Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14 (1952), 256-266; 267-274. MR 0049202 (14:135b)
  • 17. M. Gotô, Dense imbeddings of topological groups, Proc. Amer. Math. Soc. 4 (1953), 653-655. MR 0056613 (15:101c)
  • 18. M. Gotô, Dense imbeddings of locally compact connected groups, Ann. of Math. (2) 61 (1955), 154-169. MR 0065567 (16:447b)
  • 19. J. Tits, Homomorphismes ``abstraits'' de groupes de Lie, Symposia Mathematica XIII (Convegnodi Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), Academic Press, London (1974), 479-499. MR 0379749 (52:654)
  • 20. A. Borel and J. Tits, Homomorphismes ``abstraits'' de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499-571. MR 0316587 (47:5134)
  • 21. D. James, W. Waterhouse and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography; Bibliography: Abstract homomorphisms of algebraic groups, Comm. Algebra 9 (1981), no. 1, 95-114. MR 599074 (81m:20059)
  • 22. Yu. Chen, Homomorphisms from linear groups over division rings to algebraic groups, Group theory, Beijing (1984), Springer, Berlin, New York, 1986 (Lecture Notes in Math. 1185, 231-265). MR 842446 (87i:20080)
  • 23. G.M. Seitz, Abstract homomorphisms of algebraic groups, J. London Math. Soc. (2) 56 (1997), no. 1, 104-124. MR 1462829 (99b:20077)
  • 24. A.I. Shtern, The structure of homomorphisms of connected locally compact groups and compact groups, Izv. Ross. Akad. Nauk Ser. Mat., to appear.
  • 25. G.P. Hochschild, The universal representation kernel of a Lie group, Proc. Amer. Math. Soc. 11 (1960), 625-629. MR 0123640 (23:A965)
  • 26. K. Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507-558. MR 0029911 (10:679a)
  • 27. A.L.T. Paterson, Amenability, Math. Surveys and Monographs 29 (1988), Amer. Math. Soc., Providence, RI. MR 961261 (90e:43001)
  • 28. A.I. Shtern, Applications of automatic continuity results to analogs of the Freudenthal-Weil and Hochschild theorems, Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), no. 2, 203-212. MR 2656974 (2011e:22014)
  • 29. A.I. Shtern, Hochschild kernel for locally bounded finite-dimensional representations of a connected reductive Lie group, Proc. Jangjeon Math. Soc. 13 (2010), no. 2, 127-132. MR 2676681 (2011g:22013)
  • 30. A.I. Shtern, Von Neumann kernel of a connected locally compact group, revisited, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 3, 313-318. MR 2676901 (2011e:22008)
  • 31. A.I. Shtern, Homomorphic images of connected Lie groups in compact groups, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 1-6. MR 2597987 (2011e:22013)
  • 32. A.I. Shtern, Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations, Russ. J. Math. Phys. 16 (2009), no. 4, 566-567. MR 2587813 (2011a:22007)
  • 33. A.I. Shtern, Freudenthal-Weil theorem for arbitrary enbeddings of connected Lie groups in compact groups, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 2, 157-164. MR 2566913 (2011b:22008)
  • 34. K.H. Hofmann and S.A. Morris, The Lie theory of connected pro-Lie groups, European Mathematical Society, Berlin, 2007. MR 2337107 (2008h:22001)
  • 35. G. Birkhoff, Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc. 42 (1936), 883-888. MR 1563459
  • 36. M. Gotô, Faithful representations of Lie groups, Math. Japonicae I (1948), 107-119; II 1 (1950), 91-107. MR 0029919 (10:681a); MR 0038981 (12:479d)
  • 37. Harish-Chandra, On faithful representations of Lie groups, Proc. Amer. Math. Soc. 1 (1950), 205-210. MR 0034396 (11:579h)
  • 38. G.P. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco-London-Amsterdam, 1965. MR 0207883 (34:7696)
  • 39. G.P. Hochschild, Complexification of real analytic groups, Trans. Amer. Math. Soc. 125 (1966), 406-413. MR 0206141 (34:5966)
  • 40. G.P. Hochschild and G.D. Mostow, Extensions of representations of Lie groups and Lie algebras, I. Amer. J. Math. 79 (1957), 924-942. MR 0103941 (21:2703)
  • 41. D. Luminet and A. Valette, Faithful uniformly continuous representations of Lie groups, J. London Math. Soc. (2) 49 (1994), no. 1, 100-108. MR 1253015 (95c:22028)
  • 42. M. Moskowitz, A remark on faithful representations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 829-831. MR 0327979 (48:6321)
  • 43. M. Moskowitz, Faithful representations and a local property of Lie groups, Math. Z. 143 (1975), 193-198. MR 0374342 (51:10542)
  • 44. A.I. Mal$ '$tsev, Linear connected locally closed groups, Dokl. Akad. Nauk SSSR 40 (1943), no. 3, 108-110; S.R. Dokl. Acad. Sci. URSS (N.S.) 40 (1943), 87-89.
  • 45. V.S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall, Englewood Cliffs, NJ, 1974. MR 0376938 (51:13113)
  • 46. N.W. Rickert, Some properties of locally compact groups, J. Austral. Math. Soc. 7 (1967), 433-454. MR 0219656 (36:2735)
  • 47. A. Weil, L'intégration dans les groupes topologiques et ses applications, 2nd ed., Hermann, Paris, 1953.
  • 48. J. von Neumann, Almost periodic functions in a group, I, Trans. Amer. Math. Soc. 36 (1934), 445-492. MR 1501752
  • 49. A.I. Shtern, Finite-dimensional quasi-representations of connected Lie groups and Mishchenko's conjecture 13 (2007), no. 7, 85-225; English transl., J. Math. Sci. 159 (2009), no. 5, 653-751. MR 2475577 (2010c:22020)
  • 50. E. Hewitt and K.A. Ross, Abstract harmonic analysis, Springer-Verlag, Berlin-New York, 1979. MR 551496 (81k:43001)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 22E15, 22C05

Retrieve articles in all journals with MSC (2010): 22E15, 22C05


Additional Information

A. I. Shtern
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Russia
Email: ashtern@member.ams.org

DOI: https://doi.org/10.1090/S0077-1554-2012-00190-3
Keywords: Locally compact group, almost connected locally compact group, Freudenthal–Weil theorem, MAP group, semisimple locally compact group, locally bounded map.
Published electronically: January 12, 2012
Additional Notes: This work was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 08-01-000034) and the Programme of Support for Leading Scientific Schools (grant no. NSh-1562.2008.1)
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society