Enhanced Dynkin diagrams and Weyl orbits

The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow us to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in Σ. In this paper we consider only ADE root systems (i.e., systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.     

Characterization of simple Coxeter—Dynkin diagrams

Moscow Independent University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 72–75, July–September, 1995.     

On divisible weighted Dynkin diagrams and reachable elements

Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} Let e be a nilpotent element of a complex simple Lie algebra \mathfrakg \mathfrak{g} . The weighted Dynkin diagram of e, D(e) \mathcal{D}(e) , is said to be divisible if D(e)

Dynkin diagrams and short Peirce gradings of Kantor pairs

In a recent article with Oleg Smirnov, we defined short Peirce (SP) graded Kantor pairs. For any such pair P, we defined a family, parameterized by the Weyl group of type BC₂, consisting of SP-graded Kantor pairs called Weyl images of P. In this article, we classify finite dimensional simple SP-graded Kantor pairs over an algebraically closed field of characteristic 0 in terms of marked Dynkin diagrams, and we show how to compute Weyl images using these diagrams. The theory is particularly attractive for close-to-Jordan Kantor pairs (which are variations of Freudenthal triple systems), and we construct the reflections of such pairs (with nontrivial gradings) starting from Jordan pairs of matrices.     

Binary polyhedral groups and Euclidean diagrams

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.     

Dynkin diagrams of rank 20 on supersingular K3 surfaces

We classify normal supersingular K 3 surfaces Y with total Milnor number 20 in characteristic p,where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y.This paper appeared in preprint form in the home page of the first author in the year 2005.     

The Toda lattice,Dynkin diagrams,singularities and Abelian varieties

Oblatum 27-VII-1989 &; 1-III-1990     

Jordan quadratic SSM-property and its relation to copositive linear transformations on Euclidean Jordan algebras

In this paper, we introduce Jordan quadratic SSM-property and study its relation to copositive linear transformations on Euclidean Jordan algebras. In particular, we study this relationship for normal Z-transformations, Lyapunov-like transformations and cone invariant transformations.     

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matou?ek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et?al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

A property of root systems

One of the auxiliary results of a work of K. Suzuki is improved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 151–153, 1978.     

Cantor aperiodic systems and Bratteli diagrams

We prove that every Cantor aperiodic system is homeomorphic to the Vershik map acting on the space of infinite paths of an ordered Bratteli diagram and give several corollaries of this result. To cite this article: K. Medynets, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A new binary relation to compare viability of winning coalitions and its interrelationships to desirability relation and blockability relation

This paper aims to propose a new type of binary relations, called the viability relation, defined on the set of all coalitions in a simple game for a comparison of coalition influence, and to investigate its properties, especially its interrelationships to the desirability relation and the blockability relation. The viability relation is defined to compare coalitions based on their robustness over deviation of their members for complementing the inability of the desirability relation and the blockability relation to make a distinguishable comparison among winning coalitions. It is verified in this paper that the viability relation on a simple game is always transitive and is complete if and only if the simple game is S-unanimous for a coalition S. Examples show that there are no general inclusion relations among the desirability relation, the blockability relation and the viability relation. It is also verified that the viability relation and the blockability relation are complementary to each other. Specifically, the blockability relation between two coalitions is equivalent to the inversed viability relation between the complements of the two coalitions.