Zuhong Zhang

The yoga of commutators

In the present paper, we discuss some recent versions of localization methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localization, universal localization, and enhanced versions of localization-completion. Apart from the general strategic description of these methods, we state some typical technical results of conjugation calculus and commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulas, bounded width of commutators with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localization methods can be much more efficient than expected. Bibliography: 74 titles.

Generalized Commutator Formulas

Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be two-sided ideals of A, GL n (A, I) the principal congruence subgroup of level I in GL n (A), and E n (A, I) the relative elementary subgroup of level I. Using Baks localization-patching method, we prove the commutator formula which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

Multiple commutator formulas for unitary groups

Let (A,Λ) 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, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (Ii, Γi), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I0, Γ0),GU(2n, I1, Γ1), GU(2n, I2, Γ2),..., GU(2n, Im, Γm)] =[EU(2n, I0, Γ0), EU(2n, I1, Γ1), EU(2n, I2, Γ2),..., EU(2n, Im, Γm)], 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.

Generation of relative commutator subgroups in Chevalley groups

Let

Commutator width in Chevalley groups

\Phi

The Commutators of Classical Groups

be a reduced irreducible root system of rank

Relative unitary commutator calculus, and applications

\ge 2

Relative commutator calculus in Chevalley groups

, let

Multiple commutator formulas

R

The Yoga of Commutators: Further Applications

be a commutative ring and let