Amenable Bases Over Infinite Dimensional Algebras

Abstract

Given an algebra $A$ over a field $F$, a basis $mathcal{B}$ is said to be (left) amenable if the (left) $A$-algebra structure on the $F$-vector space $bigoplus_{b in mathcal{B}}Fb=F^{(mathcal{B})}cong {_AA}$ extends in a natural way to the $F$-vector space $prod_{b in mathcal{B}} Fb= F^{mathcal{B}}$. A basis $mathcal{B}$ is said to be congenial to another basis $mathcal{C}$ if infinite linear combinations of elements of the basis $mathcal{B}$ translate in a natural way to infinite linear combinations of elements of the basis $mathcal{C}$. An amenable basis is said to be simple if it is not properly congenial to any other amenable basis.The topics of this dissertation fall in four categories: begin{enumerate}item Characterizing, in terms of any fixed basis of vertices, those bases for graph magma algebras which are amenable. In addition, a complete description of all those graphs which induce graph magma algebras that have simple bases is given.We also characterize graph magma algebras that have right simple bases which are not left simple. itemDiscussing the relationship between left and right amenable basis. In particular, we introduce the notion of right (left) {it duo-amenable} algebras, those whose right (left) amenable bases are also left (right) amenable. We produce non-commutative examples of such behaviors.item Mutually congenial bases induce isomorphic basic modules. It has been shown that basic modules can indeed be non-isomorphic but it is not know whether, in general, the converse of that implication holds. We produce an example of an algebra where the converse does hold and further discuss the relation between the bases of isomorphic basic modules.item It is known that every countable algebra has an amenable basis. We explore the situation with algebras of uncountable dimension. Our discussion in this context is furthered to considerthe amenability domains of some bases.We give an example of a contrarian basis (a basis with smallest possible amenability domain) over the algebra of rational functions.