Title: Amenable bases that are not properly congenial to any others
Speaker: Rebin Muhammad (Ohio University)
Abstract: A basis B over an infinite dimensional F-algebra A is Amenable if for all r in A, the set of coordinate vectors of the family $\{rb|b \in B\} $ with respect to B is summable. When B is amenable, the A module structure of A can be extended to the vector space P consisting of the product of copies of F indexed by B. P is then said to be the basic module induced by B. Basic modules induced by bases B and C are isomorphic if B and C are mutually congenial. B is congenial to C if the coordinate vectors of the elements of B represented with respect to C are summable. If B is congenial to C but C is not congenial to B, one says that B is properly congenial to C. When B is congenial to C and C is congenial to B, we say the bases are mutually congenial. When neither B is congenial to C nor C is congenial to B, the bases are said to be discordant. An amenable basis B is simple if it is not properly congenial to any other amenable base. Al-Essa, López-Permouth and Muthana showed that the algebra A=F [x] has at least |F|pairwise discordant simple bases. We explore here the situation of polynomials with several variables in the context of studying amenability and congeniality of tensor products of algebras.
This talk reports on a collaboration with Joaquín Díaz Boils, Sergio R López-Permouth, and Ben Stanley.
