Branch rings, thinned rings, tree enveloping rings |
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| Authors: | Laurent Bartholdi |
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| Institution: | (1) Institut de mathématiques B, école Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland |
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| 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 field
% MathType!End!2!1! we contruct a
% MathType!End!2!1! which
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• is finitely generated and infinite-dimensional, but has only finitedimensional quotients;
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• has a subalgebra of finite codimension, isomorphic toM
2(k);
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• is prime;
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• has quadratic growth, and therefore Gelfand-Kirillov dimension 2;
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• is recursively presented;
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• satisfies no identity;
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• contains a transcendental, invertible element;
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• is semiprimitive if
% MathType!End!2!1! has characteristic ≠2;
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• is graded if
% MathType!End!2!1! has characteristic 2;
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• is primitive if
% MathType!End!2!1! is a non-algebraic extension of
% MathType!End!2!1!;
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• is graded nil and Jacobson radical if
% MathType!End!2!1! is an algebraic extension of
% MathType!End!2!1!.
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The author acknowledges support from TU Graz and UC Berkeley, where part of this research was conducted. |
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| Keywords: | |
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