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

A. I. Shtern

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 1, pp. 105-126.

Résumé

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.

Bibliographie

[1] Cartan É., “Sur les représentations linéaires des groupes clos”, Comment. Math. Helv., 2 (1930), 269–283 | DOI | MR | Zbl

[2] Van der Waerden B.L., “Stetigkeitssätze für halbeinfache Liesche Gruppen”, Math. Z., 36 (1933), 780–786 | DOI | MR | Zbl

[3] Lawson J.D., “Intrinsic topologies in topological lattices and semilattices”, Pacific J. Math., 44 (1973), 593–602 | MR | Zbl

[4] Comfort W.W., “Topological groups”, Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, New York, 1984, 1143–1263 | MR

[5] Pillay A., “An application of model theory to real and $p$-adic algebraic groups”, J. Algebra, 126:1 (1989), 139–146 | DOI | MR | Zbl

[6] Hofmann K.H., Morris S.A., The Structure of Compact Groups, de Gruyter, Berlin, 1998 | MR | Zbl

[7] Comfort W.W., Remus D., Szambien H., “Extending ring topologies”, J. Algebra, 232:1 (2000), 21–47 | DOI | MR | Zbl

[8] Hart J.E., Kunen K., “Bohr compactifications of non-abelian groups”, Topology Proc., 26:2 (2001/02), 593–626 | MR

[9] Izv. Math., 72:1 (2008), 169–205 | DOI | MR | Zbl

[10] Remus D., Chastnoe soobschenie, 2010

[11] Comfort W. W., Robertson L.C., “Images and quotients of $\mathrm{SO}(3,\mathbf R)$: remarks on a theorem of van der Waerden”, Rocky Mountain J. Math., 17:1 (1987), 1–13 | DOI | MR | Zbl

[12] Borel A., Essays in the History of Lie Groups and Algebraic Groups, Groups info History of Mathematics, 21, AMS, Providence, RI; London Mathematical Society, Cambridge, 2001 | MR | Zbl

[13] Schreier O., van der Waerden B.L., “Die Automorphismen der projektiven Gruppen”, Abh. Math. Sem. Hamburg, 6 (1928), 303–322 | DOI | Zbl

[14] Freudenthal H., “Die Topologie der Lieschen Gruppen als algebraisches Phänomenon I”, Ann. of Math. (2), 42 (1941), 1051–1074 | DOI | MR | Zbl

[15] Van Est W.T., “Dense imbeddings of Lie groups”, Nederl. Akad. Wetensch. Proc. Ser. A, 54, Indagationes Math., 13 (1951), 321–328 | MR | Zbl

[16] Van Est W.T., “Dense imbeddings of Lie groups II (I, II)”, Nederl. Akad. Wetensch. Proc. Ser. A, 55, Indagationes Math., 14 (1952), 255–266 ; 267–274 | MR

[17] Gotô M., “Dense imbeddings of topological groups”, Proc. Amer. Math. Soc., 4 (1953), 653–655 | DOI | MR | Zbl

[18] Gotô M., “Dense imbeddings of locally compact connected groups”, Ann. of Math. (2), 61 (1955), 154–169 | DOI | MR | Zbl

[19] Tits J., “Homomorphismes «abstraits» de groupes de Lie”, Symposia Mathematica, vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), Academic Press, London, 1974, 479–499 | MR

[20] Borel A., Tits J., “Homomorphismes «abstraits» de groupes algebriques simples”, Ann. of Math. (2), 97 (1973), 499–571 | DOI | MR | Zbl

[21] James D., Waterhouse W., Weisfeiler B., “Abstract homomorphisms of algebraic groups: problems and bibliography; Bibliography: Abstract homomorphisms of algebraic groups”, Comm. Algebra, 9:1 (1981), 95–114 | DOI | MR | Zbl

[22] Chen Yu., “Homomorphisms from linear groups over division rings to algebraic groups”, Group theory (Beijing 1984), Lecture Notes in Math., 1185, Springer, Berlin; New York, 1986, 231–265 | DOI | MR

[23] Seitz G.M., “Abstract homomorphisms of algebraic groups”, J. London Math. Soc. (2), 56:1 (1997), 104–124 | DOI | MR | Zbl

[24] Shtern A.I., “Struktura gomomorfizmov svyaznykh lokalno kompaktnykh grupp v kompaktnye gruppy”, Izvestiya RAN. Ser. matem., 75:6 (2011), 195–222 | MR

[25] Hochschild G.P., “The universal representation kernel of a Lie group”, Proc. Amer. Math. Soc., 11 (1960), 625–629 | DOI | MR | Zbl

[26] Iwasawa K., “On some types of topological groups”, Ann. of Math., 50 (1949), 507–558 | DOI | MR | Zbl

[27] Paterson A.L.T., Amenability, AMS, Providence, R.I., 1988 | MR

[28] Shtern A.I., “Applications of automatic continuity results to analogs of the Freudenthal–Weil and Hochschild theorems”, Adv. Stud. Contemp. Math. (Kyungshang), 20:2 (2010), 203–212 | MR | Zbl

[29] Shtern A.I., “Hochschild kernel for locally bounded finite-dimensional representations of a connected reductive Lie group”, Proc. Jangjeon Math. Soc., 13:2 (2010), 127–132 | MR | Zbl

[30] Shtern A.I., “Von Neumann kernel of a connected locally compact group, revisited”, Adv. Stud. Contemp. Math. (Kyungshang), 20:3 (2010), 313–318 | MR | Zbl

[31] Shtern A.I., “Homomorphic images of connected Lie groups in compact groups”, Adv. Stud. Contemp. Math. (Kyungshang), 20:1 (2010), 1–6 | MR | Zbl

[32] Shtern A.I., “Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations”, Russ. J. Math. Phys., 16:4 (2009), 566–567 | DOI | MR | Zbl

[33] Shtern A.I., “Freudenthal–Weil theorem for arbitrary embeddings of connected Lie groups in compact groups”, Adv. Stud. Contemp. Math. (Kyungshang), 19:2 (2009), 157–164 | MR | Zbl

[34] Hofmann K.H., Morris S.A., The Lie Theory of Connected Pro-Lie Groups, European Mathematical Society, Berlin, 2007 | MR

[35] Birkhoff G., “Lie groups simply isomorphic with no linear group”, Bull. Amer. Math. Soc., 42 (1936), 883–888 | DOI | MR | Zbl

[36] Gotô M., “Faithful representations of Lie groups, I”, Math. Japonicae, 1 (1948), 107–119 ; “II”, Nagoya Math. J., 1 (1950), 91–107 | MR | MR | Zbl

[37] Harish-Chandra, “On faithful representations of Lie groups”, Proc. Amer. Math. Soc., 1 (1950), 205–210 | DOI | MR | Zbl

[38] Hochschild G.P., The Structure of Lie Groups, Holden-Day; San Francisco; London; Amsterdam, 1965 | MR | Zbl

[39] Hochschild G.P., “Complexification of real analytic groups”, Trans. Amer. Math. Soc., 125 (1966), 406–413 | DOI | MR | Zbl

[40] Hochschild G.P., Mostow G.D., “Extensions of representations of Lie groups and Lie algebras. I”, Amer. J. Math., 79 (1957), 924–942 | DOI | MR | Zbl

[41] Luminet D., Valette A., “Faithful uniformly continuous representations of Lie groups”, J. London Math. Soc. (2), 49:1 (1994), 100–108 | MR | Zbl

[42] Moskowitz M., “A Remark on faithful representations”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 52 (1972), 829–831 | MR

[43] Moskowitz M., “Faithful representations and a local property of Lie groups”, Math. Z., 143 (1975), 193–198 | DOI | MR

[44] C. R. (Doklady) Acad. Sci. URSS (N. S.), 40 (1943), 87–89 | MR | Zbl

[45] Varadarajan V.S., Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974 | MR | Zbl

[46] Rickert N.W., “Some properties of locally compact groups”, J. Austral. Math. Soc., 7 (1967), 433–454 | DOI | MR | Zbl

[47] Weil A., L'intégration dans les groupes topologiques et ses applications, Act. Sc. Ind., 1145, 2éme ed., Hermann, Paris, 1953

[48] Von Neumann J., “Almost periodic functions in a group, I”, Trans. Amer. Math. Soc., 36 (1934), 445–492 | MR | Zbl

[49] Shtern A.I., “Konechnomernye kvazipredstavleniya svyaznykh grupp Li i gipoteza Mischenko”, Fundamentalnaya i prikladnaya matematika, 13:7 (2007), 85–225 ; J. Math. Sci., 159:5 (2009), 653–751 | DOI | MR | Zbl

[50] Khyuitt E., Ross K.A., Abstraktnyi garmonicheskii analiz, v. 1, Nauka, M., 1975; Hewitt E., Ross K.A., Abstract Harmonic Analysis, v. I, Springer-Verlag, Berlin; New York, 1979 | MR | Zbl