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related classifications(3)---Dynkin diagramディンキン図形20180305 のつづき・・・
The index (the n 「指数」化方式?? ) equals to the number of nodes in the diagram, the number of simple roots in a basis, the dimension of the root lattice and span of the ”root system '', the number of generators of the Coxeter group, and the rank of the Lie algebra. However, n does not equal the dimension of the defining module (a fundamental representation基礎[本?]的-表現???<'<In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a ''classical Lie group"'典型的リー群?!? is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.(これも、？けっきょく・まわりくどく??「1, 」をせつめいしているだけ ?!?!?, ) >,,>) of the Lie algebra ? the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, B _4  corresponds to s o _{2 ・ 4 + 1}   = s o _9   , which naturally acts on 9-dimensional space(Dynkin diagram? ), but has rank 4 as a Lie algebra.

The simply laced Dynkin diagramsいままでの数学的ひょうげんほうほうにないもの・として・あらたにつくりだされてもの ??? , those with no multiple edges (A, D, E) classify many further mathematical objects; see discussion at "'ADE classification '''<'~ In athematics, the ADE classification (originally A-D-E classifications) is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of π /  2 = 90° or 　2 π /3 = 120°  (single edge between the vertices). The list comprises
A _n   ,  D _n   ,  E _6   ,  E _7   ,  E _8   . 
These comprise two of the four families of Dynkin diagrams (omitting B _n  and C _n , and three of the five exceptional Dynkin diagrams (omitting F _4  and G _2  ).

This list is non-redundant if one takes n ≥ 4  for D _n.  If one extends the families to include redundant terms, one obtains the exceptional isomorphisms
D _3   ≅ A _3   , E _4   ≅ A _4   , E _5   ≅ D _5   , 
and corresponding isomorphisms of classified objects.

The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in (Arnold 1976).

The A, D, E nomenclature also yields the simply '''lacedからみあうような?? finite Coxeter groups"", by the same diagrams: in this case the 'Dynkin diagrams" exactly coincide with the ''Coxeter diagrams", as there are no multiple edges.
(規則性がない?(ちつじょがない？,)?) ようなものをつくりだし'computerｓ 『(A.I. 人工知能 )』かんり・とうかつ??・うんよう' みたいなことを・しようと・している ??? ??,
>.
   <figure--ADE classification >
The 'simply laced Dynkin diagrams'' classify diverse多種多様?!? mathematical objects.
by Wikipedia
(これらの同型は単純・半単純リー環の同型に対応し、リー群の同型にも対応する. 図形の例外同型 を、そしてリー環と付随するリー群の対応する例外同型. ・・・ 「異なる図形の間の同型に加えて、いくつかの図形は自分自身への同型すなわち「自己同型」も持つ.  図形自己同型はリー環の外部自己同型　に対応する、つまり、外部自己同型群 Out = Aut/Inn は図形の自己同型の群に等しい 」につづく？,・・ )