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 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 $\ne 2$; (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).

35 pages; small changes wrt previous version

References in corpus (4)

• Boundary representations of operator spaces, and compact rectangular matrix convex sets
• Amenability via random walks
• Dimension and randomness in groups acting on rooted trees
• A Wilson Group of Non-Uniformly Exponential Growth

Cited by in corpus (12)

• On the conjugacy problem for finite-state automorphisms of regular rooted trees
• On a conjecture of Goodearl: Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2
• Finitely constrained groups of maximal Hausdorff dimension
• Restricted Hausdorff spectra of $q$-adic automorphisms
• Extending a theorem of Herstein
• Maximal subgroups and irreducible representations of generalised multi-edge spinal groups
• Filtered Algebraic Algebras
• Rational cross-sections, bounded generation and orders on groups
• An algorithm to compute the Hausdorff dimension of regular branch groups
• Self-similar groups, automatic sequences, and unitriangular representations
• $p$-Basilica groups
• Nearly Maximal Hausdorff Dimension in Finitely Constrained Groups