In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra.
The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named.
a Dynkindiagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkindiagrams arise...
algebras, and Markov processes. The Dynkindiagram, the Dynkin system, and Dynkin's lemma are named after him. Dynkin was born into a Jewish family, living...
Aleksandr Dynkin (born 1948), Russian economist Eugene Dynkin (1924–2014), Soviet and American mathematician known for Dynkindiagram Coxeter–Dynkindiagram Dynkin...
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves...
are applied. Further, the classification scheme for root systems, by Dynkindiagrams, occurs in parts of mathematics with no overt connection to Lie theory...
obtain other Dynkindiagrams and these correspond to twisted affine Lie algebras. The attachment of an extra node to the Dynkindiagram of the corresponding...
field. In particular, the simple algebraic groups are classified by Dynkindiagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras...
mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkindiagram introduced by Satake (1960, p.109) whose configurations...
figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkindiagram, with a single ring on the end of the 4-node sequences, . The rectified...
vector spaces. Most commonly, it describes those special features of the Dynkindiagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional...
vertices). Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkindiagram, with a single ring on the end of the 1-node sequence. There are two...
generalized Dynkindiagrams. When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram). Meanwhile...
are exactly 24 Dynkindiagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkindiagrams. The complete...
the structure of the associated Dynkindiagram. In this way one may identify the automorphism group of the Dynkindiagram of G with a subgroup of Out(G)...
orientations. The 6 roots of the simple Lie group A2, represented by a Dynkindiagram , are in a regular hexagonal pattern. The two simple roots have a 120°...
{\displaystyle \beta } are in Δ {\displaystyle \Delta } , then the Dynkindiagram for Φ {\displaystyle \Phi } relative to the base Δ {\displaystyle \Delta...
where certain kinds of objects are in correspondence with simply laced Dynkindiagrams. The question of giving a common origin to these classifications, rather...
group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkindiagram, with a single ring on the end of one of the 1-node sequences. The rectified...
Gibbons (1985), or Diestel (2005). In algebra, path graphs appear as the Dynkindiagrams of type A. As such, they classify the root system of type A and the...
(nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkindiagram of . It is composed of 20 intersecting triangular faces, having five...
polytope. Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkindiagram, with a single ring on the end of one of the 2-node sequences. He also...
constructed by Ree (1960, 1961) from an exceptional automorphism of a Dynkindiagram that reverses the direction of the multiple bonds, generalizing the...
complex numbers the semisimple Lie algebras are classified by their Dynkindiagrams, of types "ABCDEFG". If L is a real simple Lie algebra, its complexification...
Global Information
