TY - JOUR

T1 - MULTIPLE COMMUTATOR FORMULAS FOR UNITARY GROUPS

AU - Hazrat, R.

AU - Vavilov, N.

AU - Zhang, Zuhong

PY - 2017/4

Y1 - 2017/4

N2 - Let (A, Lambda) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups GU(2n, A, Lambda), n >= 3. For a form ideal (I, Gamma) of the form ring (A, Lambda) we denote by EU(2n, I, Gamma) and GU(2n, I, Gamma) the relative elementary group and the principal congruence subgroup of level (I, Gamma), respectively. Now, let (I-i, Gamma(i)), i = 0, ..., m, be form ideals of the form ring (A, Lambda). The main result of the present paper is the following multiple commutator formula:[EU(2(n), I-0, G(0)), GU(2n, I-1, G(1)), GU(2n, I-2, G(2)), ..., GU(2n, I-m, Gamma(m))]=[ EU(2n, I-0, G(0)), EU(2n, I-1, Gamma(1)), EU(2n, I-2, G(2)), ..., EU(2n, I-m, Gamma(m))],which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.

AB - Let (A, Lambda) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups GU(2n, A, Lambda), n >= 3. For a form ideal (I, Gamma) of the form ring (A, Lambda) we denote by EU(2n, I, Gamma) and GU(2n, I, Gamma) the relative elementary group and the principal congruence subgroup of level (I, Gamma), respectively. Now, let (I-i, Gamma(i)), i = 0, ..., m, be form ideals of the form ring (A, Lambda). The main result of the present paper is the following multiple commutator formula:[EU(2(n), I-0, G(0)), GU(2n, I-1, G(1)), GU(2n, I-2, G(2)), ..., GU(2n, I-m, Gamma(m))]=[ EU(2n, I-0, G(0)), EU(2n, I-1, Gamma(1)), EU(2n, I-2, G(2)), ..., EU(2n, I-m, Gamma(m))],which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.

KW - ELEMENTARY SUBGROUP

KW - CHEVALLEY-GROUPS

KW - GL(N,A)

KW - K-1

KW - CLASSIFICATION

KW - STABILITY

KW - CALCULUS

KW - унитарные группы

KW - классические группы

KW - элементарные подгруппы

KW - конгруэнц-подгруппы

KW - коммутационные формулы

KW - локализационные методы

KW - квази-конечные кольца

U2 - 10.1007/s11856-017-1481-3

DO - 10.1007/s11856-017-1481-3

M3 - Article

VL - 219

SP - 287

EP - 330

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -