Branch rings, thinned rings, tree enveloping rings

By Laurent Bartholdi. 2004

66 citations recorded by the source corpus.

AbstractWe 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 $$\Bbbk $$ % MathType!End!2!1! we contruct a $$\Bbbk - algebra$$ % MathType!End!2!1! which• is finitely generated and infinite-dimensional, but has only finitedimensional quotients;• has a subalgebra of finite codimension, isomorphic toM2(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 if $$\Bbbk $$ % MathType!End!2!1! has characteristic ≠2;• is graded if $$\Bbbk $$ % MathType!End!2!1! has characteristic 2;• is primitive if $$\Bbbk $$ % MathType!End!2!1! is a non-algebraic extension of $$\mathbb{F}_2 $$ % MathType!End!2!1!;• is graded nil and Jacobson radical if $$\Bbbk $$ % MathType!End!2!1! is an algebraic extension of $$\mathbb{F}_2 $$ % MathType!End!2!1!.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb1687f8d99e05d0245456f2879ecc5ba25fe89e