Branch rings, thinned rings, tree enveloping rings

Abstract

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 fieldk % MathType!End!2!1! we contruct ak−algebra% MathType!End!2!1! which • is finitely generated and infinite-dimensional, but has only finitedimensional quotients; • has a subalgebra of finite codimension, isomorphic toM 2(k); • is prime; • has quadratic growth, and therefore Gelfand-Kirillov dimension 2; • is recursively presented; • satisfies no identity; • contains a transcendental, invertible element; • is semiprimitive ifk % MathType!End!2!1! has characteristic ≠2; • is graded ifk % MathType!End!2!1! has characteristic 2; • is primitive ifk % MathType!End!2!1! is a non-algebraic extension ofF2 % MathType!End!2!1!; • is graded nil and Jacobson radical ifk % MathType!End!2!1! is an algebraic extension ofF2% MathType!End!2!1!.