Free quotients of 𝑆𝐿₂(𝑅[𝑥])

Canadian Journal of Mathematics

We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings. If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$ . We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$ . Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$ -algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases. Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$ . If $G$ ...

Letters in Mathematical Physics, 2000

Abstract. It is shown that when q is a primitive root of unity of order not equal to 2 mod 4, A SLq 2 is a free module of ¢nite rank over the coordinate ring of the classical group SL 2 . An explicit set of generators is provided. ... Mathematics Subject Classi¢cations (2000): 17B37, ...

Journal of Algebra, 1996

2020

Jose–Rao introduced and studied the Special Unimodular Vector group SUmr(R) and EUmr(R), its Elementary Unimodular Vector subgroup. They proved that for r ≥ 2, EUmr(R) is a normal subgroup of SUmr(R). The Jose–Rao theorem says that the quotient Unimodular Vector group, SUmr(R)/EUmr(R), for r ≥ 2, is a subgroup of the orthogonal quotient group SO2(r+1)(R)/EO2(r+1)(R). The latter group is known to be nilpotent by the work of Hazrat–Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class ≤ d = dim(R). We also use the Quillen–Suslin theory, inspired by A. Bak’s method, to prove that if R = A[X ], with A a local ring, then the quotient Unimodular Vector group is abelian.

Journal of Multidisciplinary Modeling and Optimization, 2020

In this paper, we study many concepts as applications of free group, for example, presentation, rank of free group, and inverse of free group. We discussed some results about presentation concept and related it with free group. The our main result about free rank, is if G is a group, then G is free rank n if and only if G≅Zn. Also we obtained a new fact about inverse semigroup which say there is no free inverse semigroup is finitely generated as a semigroup. Moreover, we studied some results of inverse of free semigroup, These were illustrated by formulating Theorems, Lemma, Corollaries, and all of these concepts were explained through detailed examples.

Journal of the London Mathematical Society, 1997

Bulletin of the Australian Mathematical Society, 1972

In this paper it is proved that non-abelian free groups are residually [x, y \ x =1, y n =l, x = y) if and only if min{(m, k), (n, h)} is greater than 1 , and not both of (m, k) and (n, h) are 2 (where 0 is taken as greater than any natural number). The proof makes use of a result, possibly of independent interest, concerning the existence of certain automorphisms of the free group of rank two. A useful criterion which enables one to prove that non-abelian free groups are residually G for a large number of groups G is also given.

Advances in Mathematics, 2007

We classify the finite groups G such that the group of units of the integral group ring ZG has a subgroup of finite index which is a direct product of free-by-free groups.

Archiv der Mathematik, 1983

2020

Jose-Rao introduced and studied the Special Unimodular Vector group SUm_r(R) and EUm_r(R), its Elementary Unimodular Vector subgroup. They proved that for r ≥ 2, EUm_r(R) is a normal subgroup of SUm_r(R). The Jose-Rao theorem says that the quotient Unimodular Vector group, SUm_r(R)/EUm_r(R), for r ≥ 2, is a subgroup of the orthogonal quotient group SO_2(r+1)(R)/EO_2(r + 1)(R). The latter group is known to be nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class ≤ d = (R). We also use the Quillen-Suslin theory, inspired by A. Bak's method, to prove that if R = A[X], with A a local ring, then the quotient Unimodular Vector group is abelian.