Geodesic
Relative centralisers of relative subgroups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 492 (2020), pp. 10-24 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Let $R$ be an associative ring with $1$, $G=\mathrm{GL}(n, R)$ be the general linear group of degree $n\ge 3$ over $R$. In this paper we calculate the relative centralisers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal $A\unlhd R$ modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal $B\unlhd R$. Modulo congruence subgroups the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups they turned out to be quite tricky, and we could get definitive answers only over commutative rings, or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type. Bibliography: 43 titles. 
@article{ZNSL_2020_492_a1,
     author = {N. A. Vavilov and Z. Zhang},
     title = {Relative centralisers of relative subgroups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {10--24},
     year = {2020},
     volume = {492},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_492_a1/}
}
TY  - JOUR
AU  - N. A. Vavilov
AU  - Z. Zhang
TI  - Relative centralisers of relative subgroups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2020
SP  - 10
EP  - 24
VL  - 492
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2020_492_a1/
LA  - en
ID  - ZNSL_2020_492_a1
ER  - 
%0 Journal Article
%A N. A. Vavilov
%A Z. Zhang
%T Relative centralisers of relative subgroups
%J Zapiski Nauchnykh Seminarov POMI
%D 2020
%P 10-24
%V 492
%U http://geodesic.mathdoc.fr/item/ZNSL_2020_492_a1/
%G en
%F ZNSL_2020_492_a1
N. A. Vavilov; Z. Zhang. Relative centralisers of relative subgroups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 492 (2020), pp. 10-24. http://geodesic.mathdoc.fr/item/ZNSL_2020_492_a1/

[1] A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$. I. Levels and normalizers”, St. Petersburg Math. J., 23:5 (2015), 819–849 | DOI | MR

[2] A. Bak, “Non-abelian $\mathrm{K}$-theory: The nilpotent class of $\mathrm{K}_1$ and general stability”, $K$-Theory, 4 (1991), 363–397 | DOI | MR | Zbl

[3] A. Bak, R. Hazrat, N. A. Vavilov, “Localization-completion strikes again: relative $\text{\rm K}_1$ is nilpotent by abelian”, J. Pure Appl. Algebra, 213 (2009), 1075–1085 | DOI | MR | Zbl

[4] A. Bak, N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups”, Algebra Colloq., 7:2 (2000), 159–196 | DOI | MR | Zbl

[5] H. Bass, “$\mathrm{K}$-theory and stable algebra”, Inst. Hautes Études Sci. Publ. Math., 2002, no. 22, 5–60 | MR

[6] H. Bass, J. Milnor, J.-P. Serre, “Solution of the congruence subgroup problem for $\mathrm{SL}_n$ ($n\ge 3$) and $\mathrm{Sp}_{2n}$ ($n\ge 2)$”, Publ. Math. Inst. Hautes Etudes Sci., 33 (1967), 59–137 | DOI | MR | Zbl

[7] Z. I. Borewicz, N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring, containing the group of diagonal matrices”, Proc. Steklov. Inst. Math., 148 (1980), 41–54 | MR

[8] Z. I. Borewicz, N. A. Vavilov, “The distribution of subgroups in the full linear group over a commutative ring”, Proc. Steklov Inst. Math., 165 (1985), 27–46 | MR

[9] J. L. Bueso, J. Gómez-Torrencillas, A. Verschoren, Algorithmic methods in non-commutative algebra: applications to finite groups, Springer Verlag, Berlin et al., 2013 | MR

[10] A. J. Hahn, O. T. O'Meara, The classical groups and $\mathrm{K}$-theory, Springer, Berlin et al., 1989 | MR

[11] R. Hazrat, “Dimension theory and non-stable $\mathrm{K}_1$ of quadratic module”, $\mathrm{K}$-Theory, 27 (2002), 293–327 | DOI | MR

[12] R. Hazrat, A. Stepanov, N. Vavilov, Zuhong Zhang, “The yoga of commutators”, J. Math. Sci., 179:6 (2011), 662–678 | DOI | MR | Zbl

[13] R. Hazrat, A. Stepanov, N. Vavilov, Zuhong Zhang, “Commutators width in Chevalley groups”, Note di Matematica, 33:1 (2013), 139–170 | MR | Zbl

[14] R. Hazrat, N. Vavilov, “$K_1$ of Chevalley groups are nilpotent”, J. Pure Appl. Algebra, 179:1 (2003), 99–116 | DOI | MR | Zbl

[15] R. Hazrat, N. Vavilov, “Bak's work on the $K$-theory of rings”, J. K-Theory, 4:1 (2009), 1–65 | DOI | MR | Zbl

[16] R. Hazrat, N. Vavilov, Zuhong Zhang, “The commutators of classical groups”, J. Math. Sci., 222:4 (2017), 466–515 | DOI | MR | Zbl

[17] R. Hazrat, Zuhong Zhang, “Generalized commutator formula”, Commun. Algebra, 39:4 (2011), 1441–1454 | DOI | MR | Zbl

[18] R. Hazrat, Zuhong Zhang, “Multiple commutator formula”, Israel J. Math., 195 (2013), 481–505 | DOI | MR | Zbl

[19] W. van der Kallen, “A module structure on certain orbit sets of unimodular rows”, J. Pure Appl. Algebra, 57:3 (1989), 281–316 | DOI | MR | Zbl

[20] H. Marubayashi, “Primary ideal representations in non-commutative rings”, Math. J. Okayama Univ., 13:1 (1967), 1–7 | MR | Zbl

[21] A. W. Mason, “On subgroups of $\mathrm{GL}(n,A)$ which are generated by commutators. II”, J. reine angew. Math., 322 (1981), 118–135 | MR | Zbl

[22] A. W. Mason, “On non-normal subgroups of $\mathrm{GL}_n(A)$ which are normalized by elementary matrices”, Illinois J. Math., 28:1 (1984), 125–138 | DOI | MR | Zbl

[23] A. W. Mason, W. W. Stothers, “On subgroups of $\mathrm{GL}(n,A)$ which are generated by commutators”, Invent. Math., 23 (1974), 327–346 | DOI | MR | Zbl

[24] D. C. Murdoch, “Contribution to non-commutative ideal theory”, Canad. J. Math., 4:1 (1952), 43–57 | DOI | MR | Zbl

[25] O. Steinfeld, “On ideal-quotients and prime ideals”, Acta Math. Acad. Sci. Hungaricae, 4:3–4 (1953), 289–298 | DOI | MR | Zbl

[26] L. N. Vaserstein, “On the normal subgroups of the $\mathrm{GL}_n$ of a ring”, Algebraic $\mathrm{K}$-Theory, Evanston, Lecture Notes in Math., 854, Springer, Berlin et al., 1980, 454–465 | MR

[27] N. Vavilov, “Unrelativised standard commutator formula”, Zapiski Nauchnyh Seminarov POMI, 470, 2018, 38–49 | MR

[28] N. Vavilov, “Commutators of congruence subgroups in the arithmetic case”, J. Math. Sci., 479 (2019), 5–22 | MR

[29] N. Vavilov, “Towards the reverse decomposition of unipotents. II. The relative case”, J. Math. Sci., 484 (2019), 5–23 | MR

[30] N. A. Vavilov, M. R. Gavrilovich, “$\mathrm{A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm{E}_6$ and $\mathrm{E}_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672 | DOI | MR | Zbl

[31] N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718 | DOI | MR | Zbl

[32] N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921 | DOI | MR | Zbl

[33] N. A. Vavilov, S. I. Nikolenko, “$\mathrm{A}_2$-proof of structure theorems for the Chevalley group of type $\mathrm{F}_4$”, St. Petersburg Math. J., 20:4 (2009), 527–551 | DOI | MR | Zbl

[34] N. A. Vavilov, V. A. Petrov, “Overgroups of elementary symplectic groups”, St. Petersburg Math. J., 15:4 (2004), 515–543 | DOI | MR | Zbl

[35] N. A. Vavilov, V. A. Petrov, “Overgroups of $\mathrm{EO}(n,R)$”, St. Petersburg Math. J., 19:2 (2008), 167–195 | DOI | MR | Zbl

[36] N. A. Vavilov, A. V. Stepanov, “Standard commutator formula”, Vestnik St. Petersburg State Univ., ser. 1, 41:1 (2008), 5–8 | MR | Zbl

[37] N. A. Vavilov, A. V. Stepanov, “Standard commutator formula, revisited”, Vestnik St. Petersburg State Univ., ser. 1, 43:1 (2010), 12–17 | MR | Zbl

[38] N. Vavilov, Z. Zhang, “Generation of relative commutator subgroups in Chevalley groups. II”, Proc. Edinburgh Math. Soc., 63 (2020), 497–511 | DOI | MR | Zbl

[39] N. Vavilov, Z. Zhang, “Commutators of relative and unrelative elementary groups, revisited”, J. Math. Sci., 485 (2019), 58–71 | MR

[40] N. Vavilov, Z. Zhang, “Multiple commutators of elementary subgroups: end of the line”, Linear Algebra Applic., 599 (2020), 1–17 | DOI | MR | Zbl

[41] N. Vavilov, Z. Zhang, “Inclusions among commutators of elementary subgroups”, J. Algebra, 2019, 1–26 (to appear) | MR

[42] N. Vavilov, Z. Zhang, “Commutators of relative and unrelative elementary subgroups in Chevalley groups”, Edinburg Math. Soc., 2020, 1–18 (to appear) | MR

[43] N. Vavilov, Z. Zhang, “Commutators of relative and unrelative elementary unitary groups”, J. Algebra Number Theory, 2020, 1–40 (to appear) | MR