Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras (non-abelian Lie algebras without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra formula_1, if nonzero, the following conditions are equivalent: Significance. The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there  is no nonzero Lie algebra that is both solvable and semisimple. Semisimple Lie algebras have a very elegant classification, in stark contrast to solvable Lie algebras. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their root system, which are in turn classified by Dynkin diagrams. Semisimple algebras over non-algebraically closed fields can be understood in terms of those over the algebraic closure, though the classification is somewhat more intricate; see real form for the case of real semisimple Lie algebras, which were classified by Élie Cartan. Further, the representation theory of semisimple  Lie algebras is much cleaner than that for general Lie algebras. For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general. If formula_1 is semisimple, then formula_7. In particular, every linear semisimple Lie algebra is a subalgebra of formula_8, the special linear Lie algebra. The study of the structure of formula_8 constitutes an important part of the representation theory for semisimple Lie algebras. History. The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing (1888–90), though his proof  lacked rigor. His proof was made rigorous by Élie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as . Jordan decomposition. Each endomorphism "x" of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over  the algebraic closure) and nilpotent part such that "s" and "n" commute with each other. Moreover, each of "s" and "n" is a polynomial in "x". This is the Jordan decomposition of "x". The above applies to the adjoint representation formula_11 of a semisimple Lie algebra formula_1. An element "x" of formula_1 is said to be semisimple (resp. nilpotent) if formula_25 is a semisimple (resp. nilpotent) operator. If formula_26, then the abstract Jordan decomposition states that "x" can be written uniquely as: where formula_28 is semisimple, formula_29 is nilpotent and formula_30. Moreover, if formula_31 commutes with "x", then it commutes  with both formula_32 as well. The abstract Jordan decomposition factors through any representation of formula_1 in the sense that given any representation ρ, is the Jordan decomposition of ρ("x") in the endomorphism algebra of the representation space. (This is proved as a consequence of Weyl's complete reducibility theorem; see .) Structure. Let formula_1 be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of formula_1 can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition, a Cartan subalgebra (also called a maximal toral subalgebra ) formula_37 of formula_1 is a maximal subalgebra such that, for each formula_39, formula_40 is diagonalizable. As it turns out, formula_37 is abelian and so all the operators in formula_42 are simultaneously diagonalizable. For each linear functional formula_43 of formula_37, let (Note that formula_46 is the centralizer of formula_37.) Then Let formula_52 with the commutation relations formula_53; i.e., the formula_54 correspond to the standard basis of formula_49. The linear functionals in formula_56 are called the roots of formula_1 relative to formula_37. The roots span formula_59 (since if formula_60, then formula_40 is the zero operator; i.e., formula_62 is in the center, which  is zero.) Moreover, from the representation theory of formula_49, one deduces the following symmetry and integral properties of formula_56: for each formula_65, Note that formula_66 has the properties (1) formula_67 and (2) the fixed-point set is formula_68, which means that formula_66 is the reflection with respect to the hyperplane corresponding to formula_43. The above then says that formula_56 is a root system. It follows from the general theory of a root system that formula_56 contains a basis formula_73 of formula_74 such that each root is a linear combination of formula_73 with integer coefficients of the same sign; the roots  formula_76 are called simple roots. Let formula_77, etc. Then the formula_78 elements formula_79 (called Chevalley generators) generate formula_1 as a Lie algebra. Moreover, they satisfy the relations (called Serre relations): with formula_81, The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the formula_86 is a Cartan matrix). This is a theorem of Serre. In particular, two semisimple Lie algebras are isomorphic if they have the same root system. The implication of the  axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence, "all possible" semisimple Lie algebras (finite-dimensional over an algebraically closed field of characteristic zero). The Weyl group is the group of linear transformations of formula_87 generated by the formula_88's. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of formula_13 are invariant under the Weyl group. Example root space decomposition in sl(C). For formula_90 and the Cartan subalgebra formula_91 of diagonal matrices, define formula_92 by where formula_94 denotes  the diagonal matrix with formula_95 on the diagonal. Then the decomposition is given by where for the vector formula_98 in formula_99 with the standard (matrix) basis, meaning formula_98 represents the basis vector in the formula_101-th row and formula_102-th column. This decomposition of formula_13 has an associated root system: sl(C). For example, in formula_105 the decomposition is and the associated root system is sl(C). In formula_108 the decomposition is and the associated root system is given by Examples. As noted in #Structure, semisimple Lie algebras over formula_111 (or more generally an algebraically closed field of characteristic zero) are classified  by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams. Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are: The restriction formula_120 in the formula_122 family is needed because formula_123 is one-dimensional and commutative and therefore not semisimple. These Lie algebras are numbered so that "n" is the rank. Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank. For example formula_124 and  formula_125. These four families, together with five exceptions (E, E, E, F, and G), are in fact the "only" simple Lie algebras over the complex numbers. Classification. Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – A, B, C, and D – with five exceptions E, E, E, F, and G. Simple Lie algebras are classified by the connected Dynkin diagrams, shown on the right, while semisimple Lie algebras correspond to not necessarily connected Dynkin diagrams, where  each component of the diagram corresponds to a summand of the decomposition of the semisimple Lie algebra into simple Lie algebras. The classification proceeds by considering a Cartan subalgebra (see below) and its adjoint action on the Lie algebra. The root system of the action then both determines the original Lie algebra and must have a very constrained form, which can be classified by the Dynkin diagrams. See the section below describing Cartan subalgebras and root systems for more details. The classification is widely considered one of the most elegant results 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 formula_126 for A, formula_127 for B, formula_128 for C, and formula_129 for D. 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 E can also be extended down, but below E 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 formula_1 be a (finite-dimensional) semisimple Lie algebra  over an algebraically closed field of characteristic zero. Then, as in #Structure, formula_131 where formula_56 is the root system. Choose the simple roots in formula_56; a root formula_43 of formula_56 is then called positive and is denoted by formula_136 if it is a linear combination of the simple roots with non-negative integers coefficients. Let formula_137, which is a maximal solvable subalgebra of formula_1, the Borel subalgebra. Let "V" be a (possibly-infinite-dimensional) simple formula_1-module. If "V" happens to admit a formula_140-weight vector formula_141, then it is unique up to scaling and is called the highest weight  vector of "V". It is also an formula_37-weight vector and the formula_37-weight of formula_141, a linear functional of formula_37, is called the highest weight of "V". The basic yet nontrivial facts then are (1) to each linear functional formula_146, there exists a simple formula_1-module formula_148 having formula_149 as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between formula_59 and the set of the equivalence classes of simple formula_1-modules admitting a Borel-weight vector. For applications, one is often interested in a finite-dimensional  simple formula_1-module (a finite-dimensional irreducible representation). This is especially the case when formula_1 is the Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber formula_154, we mean the convex cone formula_155 where formula_156 is a unique vector such that formula_157. The criterion then reads: A linear functional formula_149 satisfying the above equivalent condition is called a dominant integral weight. Hence, in summary, there  exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple formula_1-modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character formula. The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra formula_1 is completely reducible; i.e., it is a direct sum of simple formula_1-modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra. Real semisimple Lie algebra . For a semisimple Lie algebra over a field that has characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic zero. But over the field of real numbers, there are still the structure results. Let formula_1 be a finite-dimensional real semisimple Lie algebra and formula_168 the complexification of it (which is again semisimple). The real Lie algebra formula_1 is called a real form of formula_170. A real form is called a compact form if the Killing form on it is negative-definite; it is  necessarily the Lie algebra of a compact Lie group (hence, the name). Compact case. Suppose formula_1 is a compact form and formula_172 a maximal abelian subspace. One can show (for example, from the fact formula_1 is the Lie algebra of a compact Lie group) that formula_42 consists of skew-Hermitian matrices, diagonalizable over formula_111 with imaginary eigenvalues. Hence, formula_176 is a Cartan subalgebra of formula_170 and there results in the root space decomposition (cf. #Structure) where each formula_179 is real-valued on formula_180; thus, can be identified with a real-linear functional on the real vector space formula_180. For example , let formula_182 and take formula_172 the subspace of all diagonal matrices. Note formula_184. Let formula_185 be the linear functional on formula_186 given by formula_187 for formula_188. Then for each formula_189, where formula_191 is the matrix that has 1 on the formula_192-th spot and zero elsewhere. Hence, each root formula_43 is of the form formula_194 and the root space decomposition is the decomposition of matrices: Noncompact case. Suppose formula_1 is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a Cartan involution formula_197 and let formula_198 be the eigenspace  decomposition of formula_197, where formula_200 are the eigenspaces for 1 and -1, respectively. For example, if formula_201 and formula_197 the negative transpose, then formula_203. Let formula_204 be a maximal abelian subspace. Now, formula_205 consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in formula_206 are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the restricted root space decomposition): where Moreover, formula_56 is a root system but not necessarily reduced one (i.e., it can happen formula_214 are both roots). The case of sl(n ,C). If formula_215, then formula_91 may be taken to be the diagonal subalgebra of formula_13, consisting of diagonal matrices whose diagonal entries sum to zero. Since formula_91 has dimension formula_219, we see that formula_220 has rank formula_219. The root vectors formula_222 in this case may be taken to be the matrices formula_223 with formula_224, where formula_223 is the matrix with a 1 in the formula_226 spot and zeros elsewhere. If formula_227 is a diagonal matrix with diagonal entries formula_228, then we have Thus, the roots for formula_230 are the linear functionals formula_231 given by After identifying formula_91 with its dual , the roots become the vectors formula_234 in the space of formula_29-tuples that sum to zero. This is the root system known as formula_236 in the conventional labeling. The reflection associated to the root formula_231 acts on formula_91 by transposing the formula_101 and formula_102 diagonal entries. The Weyl group is then just the permutation group on formula_29 elements, acting by permuting the diagonal entries of matrices in formula_91. Generalizations. Semisimple Lie algebras admit certain generalizations. Firstly, many statements that are true for semisimple Lie algebras are true more generally for reductive Lie algebras. Abstractly, a reductive Lie algebra is one  whose adjoint representation is completely reducible, while concretely, a reductive Lie algebra is a direct sum of a semisimple Lie algebra and an abelian Lie algebra; for example, formula_243 is semisimple, and formula_244 is reductive. Many properties of semisimple Lie algebras depend only on reducibility. Many properties of complex semisimple/reductive Lie algebras are true not only for semisimple/reductive Lie algebras over algebraically closed fields, but more generally for split semisimple/reductive Lie algebras over other fields: semisimple/reductive Lie algebras over algebraically closed fields are always split, but over other fields this is not always the case. Split  Lie algebras have essentially the same representation theory as semsimple Lie algebras over algebraically closed fields, for instance, the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in , for instance, which classifies representations of split semisimple/reductive Lie algebras. Semisimple and reductive groups. A connected Lie group is called semisimple if its Lie algebra is a semisimple Lie algebra, i.e. a direct sum of simple Lie algebras. It is called reductive if its Lie algebra is a direct sum of simple and trivial (one-dimensional) Lie algebras . Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics. For example, the group formula_245 of symmetries of an "n"-dimensional real vector space (equivalently, the group of invertible matrices) is reductive.