Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations
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A. I. Shtern
Translated by: G. G. Gould - Trans. Moscow Math. Soc. 2011, 79-95
- DOI: https://doi.org/10.1090/S0077-1554-2012-00190-3
- Published electronically: January 12, 2012
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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
- Elie Cartan, Sur les représentations linéaires des groupes clos, Comment. Math. Helv. 2 (1930), no. 1, 269–283 (French). MR 1509418, DOI 10.1007/BF01214464
- B. L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933), no. 1, 780–786 (German). MR 1545369, DOI 10.1007/BF01188647
- Jimmie D. Lawson, Intrinsic topologies in topological lattices and semilattices, Pacific J. Math. 44 (1973), 593–602. MR 318031
- W. W. Comfort, Topological groups, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1143–1263. MR 776643
- Anand Pillay, An application of model theory to real and $p$-adic algebraic groups, J. Algebra 126 (1989), no. 1, 139–146. MR 1023289, DOI 10.1016/0021-8693(89)90323-2
- Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, De Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. A primer for the student—a handbook for the expert. MR 1646190
- W. W. Comfort, Dieter Remus, and Horst Szambien, Extending ring topologies, J. Algebra 232 (2000), no. 1, 21–47. MR 1783911, DOI 10.1006/jabr.2000.8382
- Joan E. Hart and Kenneth Kunen, Bohr compactifications of non-abelian groups, Proceedings of the 16th Summer Conference on General Topology and its Applications (New York), 2001/02, pp. 593–626. MR 2032839
- 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 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 1, 169–205. MR 2394977, DOI 10.1070/IM2008v072n01ABEH002397
- D. Remus, Private communication (2010).
- W. W. Comfort and Lewis C. Robertson, Images and quotients of $\textrm {SO}(3,\textbf {R})$: remarks on a theorem of van der Waerden, Rocky Mountain J. Math. 17 (1987), no. 1, 1–13. MR 882879, DOI 10.1216/RMJ-1987-17-1-1
- Armand Borel, Essays in the history of Lie groups and algebraic groups, History of Mathematics, vol. 21, American Mathematical Society, Providence, RI; London Mathematical Society, Cambridge, 2001. MR 1847105, DOI 10.1090/hmath/021
- O. Schreier and B. L. van der Waerden, Die Automorphismen der projektiven Gruppen, Abh. Math. Sem. Hamburg 6 (1928), 303–322.
- Hans Freudenthal, Die Topologie der Lieschen Gruppen als algebraisches Phänomen. I, Ann. of Math. (2) 42 (1941), 1051–1074 (German). MR 5740, DOI 10.2307/1970461
- W. T. van Est, Dense imbeddings of Lie groups, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 321–328. MR 0044530
- W. T. van Est, Dense imbeddings of Lie groups. II(I,II), Nederl. Akad. Wetensch. Proc. Ser. A. 55=Indagationes Math. 14 (1952), 255–266, 267–274. MR 0049202
- Morikuni Goto, Dense imbedding of topological groups, Proc. Amer. Math. Soc. 4 (1953), 653–655. MR 56613, DOI 10.1090/S0002-9939-1953-0056613-3
- Morikuni Goto, Dense imbeddings of locally compact connected groups, Ann. of Math. (2) 61 (1955), 154–169. MR 65567, DOI 10.2307/1969626
- J. Tits, Homorphismes “abstraits” de groupes de Lie, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani & Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 479–499 (French). MR 0379749
- Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI 10.2307/1970833
- D. James, W. Waterhouse, and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography, Comm. Algebra 9 (1981), no. 1, 95–114. MR 599074, DOI 10.1080/00927878108822565
- Yu Chen, Homomorphisms from linear groups over division rings to algebraic groups, Group theory, Beijing 1984, Lecture Notes in Math., vol. 1185, Springer, Berlin, 1986, pp. 231–265. MR 842446, DOI 10.1007/BFb0076177
- Gary M. Seitz, Abstract homomorphisms of algebraic groups, J. London Math. Soc. (2) 56 (1997), no. 1, 104–124. MR 1462829, DOI 10.1112/S0024610797005176
- A. I. Shtern, The structure of homomorphisms of connected locally compact groups and compact groups, Izv. Ross. Akad. Nauk Ser. Mat., to appear.
- G. Hochschild, The universal representation kernel of a Lie group, Proc. Amer. Math. Soc. 11 (1960), 625–629. MR 123640, DOI 10.1090/S0002-9939-1960-0123640-0
- Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- Alexander 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
- Alexander 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
- Alexander 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
- Alexander I. Shtern, Homomorphic images of connected Lie groups in compact groups, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 1–6. MR 2597987
- 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, DOI 10.1134/S1061920809040116
- Alexander I. Shtern, Freudenthal-Weil theorem for arbitrary embeddings of connected Lie groups in compact groups, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 2, 157–164. MR 2566913
- Karl H. Hofmann and Sidney A. Morris, The Lie theory of connected pro-Lie groups, EMS Tracts in Mathematics, vol. 2, European Mathematical Society (EMS), Zürich, 2007. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. MR 2337107, DOI 10.4171/032
- Garrett Birkhoff, Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc. 42 (1936), no. 12, 883–888. MR 1563459, DOI 10.1090/S0002-9904-1936-06457-8
- Morikuni Gotô, Faithful representations of Lie groups. I, Math. Japon. 1 (1948), 107–119. MR 29919
- Harish-Chandra, On faithful representations of Lie groups, Proc. Amer. Math. Soc. 1 (1950), 205–210. MR 34396, DOI 10.1090/S0002-9939-1950-0034396-8
- G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
- G. Hochschild, Complexification of real analytic groups, Trans. Amer. Math. Soc. 125 (1966), 406–413. MR 206141, DOI 10.1090/S0002-9947-1966-0206141-0
- G. Hochschild and G. D. Mostow, Extensions of representations of Lie groups and Lie algebras. I, Amer. J. Math. 79 (1957), 924–942. MR 103941, DOI 10.2307/2372443
- Denis Luminet and Alain Valette, Faithful uniformly continuous representations of Lie groups, J. London Math. Soc. (2) 49 (1994), no. 1, 100–108. MR 1253015, DOI 10.1112/jlms/49.1.100
- Martin Moskowitz, A remark on faithful representations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 52 (1972), 829–831 (1973) (English, with Italian summary). MR 327979
- Martin Moskowitz, Faithful representations and a local property of Lie groups, Math. Z. 143 (1975), no. 2, 193–198. MR 374342, DOI 10.1007/BF01187063
- 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.
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938
- Neil W. Rickert, Some properties of locally compact groups, J. Austral. Math. Soc. 7 (1967), 433–454. MR 0219656
- A. Weil, L’intégration dans les groupes topologiques et ses applications, 2nd ed., Hermann, Paris, 1953.
- J. v. Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. 36 (1934), no. 3, 445–492. MR 1501752, DOI 10.1090/S0002-9947-1934-1501752-3
- A. I. Shtern, Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko’s conjecture, Fundam. Prikl. Mat. 13 (2007), no. 7, 85–225 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 159 (2009), no. 5, 653–751. MR 2475577, DOI 10.1007/s10958-009-9466-3
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Bibliographic Information
- A. I. Shtern
- Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Russia
- 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)
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2011, 79-95
- MSC (2010): Primary 22E15, 22C05
- DOI: https://doi.org/10.1090/S0077-1554-2012-00190-3
- MathSciNet review: 3184813