in mathematics – a brief list of axioms yields, via a relatively short proof, a complete but non-trivial classification with surprising structure. This should be compared to the classification of finite simple groups, which is significantly more complicated. The enumeration of the four families is non-redundant and consists only of simple algebras if for An, for Bn, for Cn, and for Dn. If one starts numbering lower, the enumeration is redundant, and one has exceptional isomorphisms between simple Lie algebras, which are reflected in isomorphisms of Dynkin diagrams; the En can also be extended down, but below E6 are isomorphic to other, non-exceptional algebras. Over a non-algebraically closed field, the classification is more complicated – one classifies simple Lie algebras over the algebraic closure, then for each of these, one classifies simple Lie algebras over the original field which have this form (over the closure). For example, to classify simple real Lie algebras, one classifies real Lie algebras with a given complexification, which are known as real forms of the complex Lie algebra; this can be done by Satake diagrams, which are Dynkin diagrams with additional data ("decorations"). Representation theory of semisimple Lie algebras Let be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in #Structure, where is the root system. Choose the simple roots in ; a root of is then called positive and is denoted by if it is a linear combination of the simple roots with non-negative integer coefficients. Let , which is a maximal solvable subalgebra of , the Borel subalgebra. Let V be a (possibly-infinite-dimensional) simple -module. If V happens to admit a -weight vector , then it is unique up to scaling and is called the highest weight vector of V. It is also an -weight vector and the -weight of , a linear functional of , is called the highest weight of V. The basic yet nontrivial facts then are (1) to each linear functional , there exists a simple -module having as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between and the set of the equivalence classes of simple -modules admitting a Borel-weight vector. For applications, one is often interested in a finite-dimensional simple -module (a finite-dimensional irreducible representation). This is especially the case when is the Lie algebra of a