TY - JOUR

T1 - Normal structure of isotropic reductive groups over rings

AU - Степанов, Алексей Владимирович

AU - Ставрова, Анастасия Константиновна

PY - 2024/10/15

Y1 - 2024/10/15

N2 - The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the isotropic rank of G is at least 2 and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)⩽…⩽C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R)→G(R/q). The main ingredients of the proof are the “level computation” by the first author and the generic element method developed by the second author.

AB - The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the isotropic rank of G is at least 2 and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)⩽…⩽C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R)→G(R/q). The main ingredients of the proof are the “level computation” by the first author and the generic element method developed by the second author.

KW - Congruence subgroup

KW - Elementary subgroup

KW - Generic element

KW - Isotropic reductive groups

KW - Normal structure

KW - Parabolic subgroup

KW - Unipotent element

KW - Universal localization

UR - https://www.mendeley.com/catalogue/89bc1058-8a8e-3f12-8a02-c6d6a437bc4c/

U2 - 10.1016/j.jalgebra.2022.11.014

DO - 10.1016/j.jalgebra.2022.11.014

M3 - Article

VL - 656

SP - 486

EP - 515

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -