We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by Chevalley on general groups. Apart from proving some properties of Cartan subgroups, we show that the Cartan subgroups of the quotient groups are precisely the images of Cartan subgroups of the ambient group. We establish the so-called “Levi” decomposition of Cartan subgroups which extends Wüstner’s decomposition theorem and our earlier results for Lie groups. We also show that the centraliser of any maximal torus of the radical is connected and its Cartan subgroups are also Cartan subgroups of the ambient group; moreover, every Cartan subgroup arises this way. We prove that Cartan subalgebras defined by Hofmann and Morris in pro-Lie algebras are the same as those corresponding to Cartan subgroups in case of pro-Lie algebras of connected locally compact groups, and that they are nilpotent. We characterise density of the image of a power map in a connected locally compact group in terms of its surjectivity on all Cartan subgroups, and show that weak exponentiality of the group is equivalent to the condition that all its Cartan subgroups are connected.
Accepted:
Published online:
Keywords: Connected locally compact groups, Cartan subgroups, Cartan subalgebras, power maps
Mandal, Arunava 1 ; Shah, Riddhi 2
Mandal, Arunava; Shah, Riddhi. Cartan Subgroups in Connected Locally Compact Groups. Journal of Lie Theory, Volume 36 (2026) no. 2-3, pp. 487-516. doi: 10.5802/jolt.1437
@article{10_5802_jolt_1437,
author = {Mandal, Arunava and Shah, Riddhi},
title = {Cartan {Subgroups} in {Connected} {Locally} {Compact} {Groups}},
journal = {Journal of Lie Theory},
pages = {487--516},
year = {2026},
publisher = {XXXX},
volume = {36},
number = {2-3},
doi = {10.5802/jolt.1437},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1437/}
}
TY - JOUR AU - Mandal, Arunava AU - Shah, Riddhi TI - Cartan Subgroups in Connected Locally Compact Groups JO - Journal of Lie Theory PY - 2026 SP - 487 EP - 516 VL - 36 IS - 2-3 PB - XXXX UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1437/ DO - 10.5802/jolt.1437 LA - en ID - 10_5802_jolt_1437 ER -
[1] On the density of images of the power maps in Lie groups, Arch. Math., Volume 110 (2018) no. 2, pp. 115-130 | DOI | Zbl
[2] Groupes linéaires algébriques, Ann. Math. (2), Volume 64 (1956), pp. 20-82 | DOI | Zbl | MR
[3] Théorie des Groupes de Lie III. Tome III. Théorémes Généraux sur les Algébres de Lie, Actualités Scientifiques et Industrielles, 1226, Hermann, 1955 | Zbl | MR
[4] On the surjectivity of the power maps of a class of solvable groups, J. Group Theory, Volume 20 (2017) no. 6, pp. 1089-1101 | DOI | Zbl | MR
[5] The structure of locally compact groups, Duke Math. J., Volume 18 (1951), pp. 85-104 | DOI | Zbl | MR
[6] Cartan subgroups of a Lie group, J. Math. Soc. Japan, Volume 32 (1980), pp. 251-262 | DOI | Zbl | MR
[7] Probability measures on locally compact groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 94, Springer, 1977 | Zbl | DOI | MR
[8] The Lie Theory of Connected Pro-Lie Groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups, EMS Tracts in Mathematics, 2, European Mathematical Society, 2007 | DOI | Zbl | MR
[9] On the density of the image of the exponential function, Math. Ann., Volume 234 (1978), pp. 263-273 | DOI | Zbl | MR
[10] The Structure of Compact Groups, De Gruyter Studies in Mathematics, 25, Walter de Gruyter, 1998 | Zbl | MR
[11] The compact generation of closed subgroups of locally compact groups, J. Group Theory, Volume 12 (2009) no. 4, pp. 555-559 | DOI | Zbl | MR
[12] Near-Cartan algebras and groups, Semin. Sophus Lie, Volume 2 (1992) no. 2, pp. 135-151 | Zbl | MR
[13] Abstract Harmonic Analysis Vol. 1: Structure of topological groups; integration theory; group representations, Grundlehren der Mathematischen Wissenschaften, 115, Springer, 1979 | Zbl | DOI | MR
[14] On some types of topological groups, Ann. Math. (2), Volume 50 (1949), pp. 507-558 | DOI | Zbl
[15] Lie algebras of locally compact groups, Pac. J. Math., Volume 7 (1957), pp. 1145-1162 | DOI | Zbl | MR
[16] Dense images of the power maps in Lie groups and minimal parabolic subgroups, New York J. Math., Volume 24 (2018), pp. 201-209 | Zbl | MR
[17] Dense images of the power maps for a disconnected real algebraic group, J. Group Theory, Volume 24 (2021) no. 5, pp. 973-985 | DOI | Zbl | MR
[18] On the decomposition of an (${L}$)-group, J. Math. Soc. Japan, Volume 1 (1950), pp. 264-274 | DOI | Zbl | MR
[19] On $n$th roots and infinitely divisible elements in a connected Lie group, Math. Proc. Camb. Philos. Soc., Volume 89 (1981), pp. 293-299 | DOI | Zbl | MR
[20] Group actions and power maps for groups over non-Archimedean local fields (2021) | arXiv | Zbl
[21] The structure of Cartan subgroups in Lie groups, Math. Z., Volume 299 (2021) no. 3-4, pp. 1587-1606 | DOI | Zbl | MR
[22] Weakly exponential Lie groups, J. Algebra, Volume 179 (1996) no. 2, pp. 331-361 | DOI | Zbl | MR
[23] Cartan subgroups and lattices in semi-simple groups, Ann. Math. (2), Volume 96 (1972), pp. 296-317 | DOI | Zbl | MR
[24] A normal subgroup of a semisimple Lie group is closed, Proc. Am. Math. Soc., Volume 32 (1972), pp. 632-633 | DOI | Zbl | MR
[25] On Cartan subgroups of linear groups, J. Sci. Hiroshima Univ., Ser. A-I, Volume 25 (1961), pp. 63-93 | Zbl | MR
[26] Generic subgroups of Lie groups, Topology, Volume 41 (2002) no. 1, pp. 163-181 | DOI | Zbl | MR
[27] A generalization of the Jordan decomposition, Forum Math., Volume 15 (2003) no. 3, pp. 395-408 | DOI | Zbl | MR
[28] On the surjectivity of the exponential function of solvable Lie groups, Math. Nachr., Volume 192 (1998), pp. 255-266 | DOI | Zbl | MR
[29] A generalization of a theorem of Gleason, Ann. Math. (2), Volume 58 (1953), pp. 351-365 | DOI | Zbl
Cited by Sources: