In this section we review our basic algebraic concepts and notation. For more information, look up a basic textbook in linear algebra [1]. 1.1 Logic and sets All mathematics is based on an underlying understanding of logic. It starts of course with proposition logic, but predicate logic is the most relevant. We borrow from it its basic notation, and its basic modes of reasoning. Examples of (so...

This paper studies three classes of discrete sets X in R n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X ? X. A nitely generated Delone set is one such that the abelian group X ? X] generated by X ? X is nitely generated, so that X ? X] is a lattice or a quasilattice. For such sets the abelian group X] is nitely generated...

In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with respect to the real case, a matrix of order n could have more than n eigenvalues (the set of intervals is not factorial). We consider a notion of central eigen...

We survey three recent developments in algebraic combinatorics. The first is the theory of cluster algebras and the Laurent phenomenon of Sergey Fomin and Andrei Zelevinsky. The second is the construction of toric Schur functions and their application to computing three-point Gromov-Witten invariants, by Alexander Postnikov. The third development is the construction of intersection cohomology f...

a polynomial   1 2 ( , , , ) n f x x x is called multilinear if it is homogeneous and linear in every one of its variables. in the present paper our objective is to prove the following result: let   r be a prime k-algebra over a commutative ring   k with unity and let 1 2 ( , , , ) n f x x x be a multilinear polynomial over k. suppose that   d is a nonzero derivation on r such that ...

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g,A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In many examples A is the universal enveloping algebra of g. Our description of the algebra (g,A)(F)...

The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F -loop algebra (g, A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In most examples A is the universal enveloping algebra of g. Our description of the loop algebra has a str...