Branch rings, thinned rings, tree enveloping rings |
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Authors: | Laurent Bartholdi |
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Affiliation: | (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 – | • 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 if % MathType!End!2!1! has characteristic ≠2; | – | • is graded if % MathType!End!2!1! has characteristic 2; | – | • is primitive if % MathType!End!2!1! is a non-algebraic extension of % MathType!End!2!1!; | – | • is graded nil and Jacobson radical if % MathType!End!2!1! is an algebraic extension of % MathType!End!2!1!. | The author acknowledges support from TU Graz and UC Berkeley, where part of this research was conducted. |
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