18 research outputs found
Branch Rings, Thinned Rings, Tree Enveloping Rings
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 ;
(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 ;
(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).Comment: 35 pages; small changes wrt previous versio
Open Problems on Central Simple Algebras
We provide a survey of past research and a list of open problems regarding
central simple algebras and the Brauer group over a field, intended both for
experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered,
compared to v
Generators of central simple p-algebras of degree 3
We discuss standard pairs of generators of cyclic division p-algebras of degree p, and prove for p = 3 that any two Artin-Schreier elements are connected by a chain of standard pairs. This result has immediate applications to the presentations of such algebras
Hashing into Hessian curves
We describe a hashing function from the elements of the finite field double-struck Fq into points on a Hessian curve. Our function features the uniform and smaller size for the cardinalities of almost all fibers compared with the other known hashing functions for elliptic curves. For ordinary Hessian curves, this function is 2:1 for almost all points. More precisely, for odd q, the cardinality of the image set of the function is exactly given by (q + i + 2)/2 for some i = - 1,1. Next, we present an injective hashing function from the elements of ℤm into points on a Hessian curve over double-struck Fq with odd q and m = (q + i)/2 for some i = - 1,1,3.12 page(s