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The Cartan classification of semisimple algebras relies on the observation that the number of ways that one can draw in the *r*-dimensional Euclidean root space a so-called root diagram is very limited. For rank one there is only one root diagram possible, the corresponding algebra being *A* _{1} ≅ *B* _{1} ≅ *C* _{1} in Cartan’s notation. For rank 2 there are four different root diagrams possible, corresponding to the algebras *A* _{2}, *B* _{2} ≅ *C* _{2}, *D* _{2} ≅ *A* _{1}⊕*A* _{1} and *G* _{2}. By generalizing these one obtains the infinite series *A _{n}*,

*B*,

_{n}*C*and

_{n}*D*of rank

_{n}*n*algebras, which are the classical algebras.

*G*

_{2}cannot be generalized but other exceptional algebras exist, namely

*F*

_{4}and

*E*

_{6},

*E*

_{7}and

*E*

_{8}.

*Keywords: *
Cartan classification;
root diagrams;
classical algebras;
exceptional algebras

*Chapter.*
*3923 words.*
*Illustrated.*

*Subjects: *
Mathematical and Statistical Physics

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