Abstract
For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.
Highlights
For any complex semisimple Lie algebra, there are a number of mathematical objects that are traditionally attached to it, and which determine it to some extent
The most widely used mathematical objects are: the Dynkin diagram, the Cartan matrix, the system of positive roots, the system of simple roots, the Weyl group, the universal enveloping algebra, etc. These objects have proved their usefulness in dealing with complex semisimple Lie algebras, and most of them have been generalized in order to deal with the new classes of mathematical structures, such as Kac-Moody algebras, superalgebras, quantum groups and Coxeter systems
Loutsiouk where aij is an element of the Cartan matrix A defined by the system Π of simple roots, and ei is an element of the standard basis of n, which we identify with the simple root αi
Summary
For any complex semisimple Lie algebra, there are a number of mathematical objects that are traditionally attached to it, and which determine it to some extent. The most widely used mathematical objects are: the Dynkin diagram, the Cartan matrix, the system of positive roots, the system of simple roots, the Weyl group, the universal enveloping algebra, etc. These objects have proved their usefulness in dealing with complex semisimple Lie algebras, and most of them have been generalized in order to deal with the new classes of mathematical structures, such as Kac-Moody algebras, superalgebras, quantum groups and Coxeter systems. Two alternative mathematical objects are defined for any complex semisimple Lie algebra G. These objects are ellipsoids in the real linear space n , where n is the rank of G. We assume the linear space n to be partially ordered as follows: x ≤ y if and only if for any 1 ≤ i ≤ n we have xi ≤ yi
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