Branch Rings, Thinned Rings, Tree Enveloping Rings

We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which (1) is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; (2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$; (3) is prime; (4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2; (5) is recursively presented; (6) satisfies no identity; (7) contains a transcendental, invertible element; (8) is semiprimitive if k has characteristic $\neq2$; (9) is graded if k has characteristic 2; (10) is primitive if k is a non-algebraic extension of GF(2); (11) is graded nil and Jacobson radical if k is an algebraic extension of GF(2).Comment: 35 pages; small changes wrt previous versio

Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.Comment: 21 pages, 6 figures and 6 tables, submitted to Prog. Theor. Phy

On pairs of matrices generating matrix rings and their presentations

AbstractLet Mn(Z) be the ring of n-by-n matrices with integral entries, and n⩾2. This paper studies the set Gn(Z) of pairs (A,B)∈Mn(Z)2 generating Mn(Z) as a ring. We use several presentations of Mn(Z) with generators X=∑i=1nEi+1,i and Y=E11 to obtain the following consequences.(1)Let k⩾1. The following rings have presentations with 2 generators and finitely many relations:(a)⊕j=1kMmj(Q) for any m1,…,mk⩾2.(b)⊕j=1kMnj(Z), where n1,…,nk⩾2, and the same ni is repeated no more than three times.(2)Let D be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for X and Y to describe all representations of the ring Mn(D) into Mm(D) for m⩾n.(3)We obtain information about the asymptotic density of Gn(F) in Mn(F)2 over different fields, and over the integers

THE GROUP OF AUTOMORPHISMS OF A 3-GENERATED 2-GROUP OF INTERMEDIATE GROWTH

We decidate this paper to John Rhodes on his 65th birthday. Abstract. The automorphism group of a 3-generated 2-group G of intermediate growth is determined and it is shown that the outer group of automorphisms of G to be an elementary abelian 3- group of infinite rank. 1