Multiple Commutator Formulas

TL;DR

This paper proves a generalized multiple commutator formula for principal congruence and elementary subgroups in quasi-finite algebras, extending classical results in algebraic K-theory and group theory.

Contribution

It establishes a broad generalization of standard commutator formulas for multiple levels of ideals in quasi-finite algebras.

Findings

Proves a multiple commutator formula for quasi-finite algebras.

Generalizes classical commutator formulas to multiple ideals.

Extends understanding of subgroup interactions in algebraic structures.

Abstract

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \GL_n(A,I_i) the principal congruence subgroup of level I_i in GL_n(A) and E_n(A,I_i) be the relative elementary subgroup of level I_i. We prove a multiple commutator formula [E_n(A,I_0),\GL_n(A,I_1),& \GL_n(A, I_2),..., \GL_n(A, I_m)] = [E_n(A,I_0),E_n(A,I_1),E_n(A, I_2),..., E_n(A, I_m)], which is a broad generalization of the standard commutator formulas.