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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
is semisimple if it is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra 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 \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
, κ(x,y) = tr(ad(''x'')ad(''y'')), is
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...
; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
(maximal solvable ideal) of \mathfrak g is zero.


Significance

The significance of semisimplicity comes firstly from the
Levi decomposition In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a ...
, 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 algebra In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consist ...
s. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
, which are in turn classified by
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s. 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
. 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 \mathfrak g is semisimple, then \mathfrak g = mathfrak g, \mathfrak g/math>. In particular, every linear semisimple Lie algebra is a subalgebra of \mathfrak, the special linear Lie algebra. The study of the structure of \mathfrak 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
(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 .


Basic properties

*Every ideal, quotient and product of semisimple Lie algebras is again semisimple. *The center of a semisimple Lie algebra \mathfrak g is trivial (since the center is an abelian ideal). In other words, the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
\operatorname is injective. Moreover, the image turns out to be \operatorname(\mathfrak g) of derivations on \mathfrak. Hence, \operatorname: \mathfrak \overset\to \operatorname(\mathfrak g) is an isomorphism. (This is a special case of Whitehead's lemma.) *As the adjoint representation is injective, a semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space ( Ado's theorem), although not necessarily via the adjoint representation. But in practice, such ambiguity rarely occurs. *If \mathfrak g is a semisimple Lie algebra, then \mathfrak g = mathfrak g, \mathfrak g/math> (because \mathfrak g/ mathfrak g, \mathfrak g/math> is semisimple and abelian). *A finite-dimensional Lie algebra \mathfrak g over a field ''k'' of characteristic zero is semisimple if and only if the base extension \mathfrak \otimes_k F is semisimple for each field extension F \supset k. Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple.


Jordan decomposition

Each
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
''x'' of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
(i.e., diagonalizable over the algebraic closure) and
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
part :x=s+n\ 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 In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
\operatorname of a semisimple Lie algebra \mathfrak g. An element ''x'' of \mathfrak g is said to be semisimple (resp. nilpotent) if \operatorname(x) is a semisimple (resp. nilpotent) operator. If x\in\mathfrak g, then the abstract Jordan decomposition states that ''x'' can be written uniquely as: :x = s + n where s is semisimple, n is nilpotent and
, n The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= 0. Moreover, if y \in \mathfrak g commutes with ''x'', then it commutes with both s, n as well. The abstract Jordan decomposition factors through any representation of \mathfrak g in the sense that given any representation ρ, :\rho(x) = \rho(s) + \rho(n)\, 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 Weyl's theorem on complete reducibility#Application: preservation of Jordan decomposition.)


Structure

Let \mathfrak g be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of \mathfrak g can be described by an
adjoint action In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
of a certain distinguished subalgebra on it, a
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
. By definition, a
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
(also called a maximal toral subalgebra) \mathfrak h of \mathfrak g is a maximal subalgebra such that, for each h \in \mathfrak h, \operatorname(h) is
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
. As it turns out, \mathfrak h is abelian and so all the operators in \operatorname(\mathfrak h) are
simultaneously diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique. ...
. For each linear functional \alpha of \mathfrak h, let :\mathfrak_ = \. (Note that \mathfrak_0 is the
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
of \mathfrak h.) Then (The most difficult item to show is \dim \mathfrak_ = 1. The standard proofs all use some facts in the representation theory of \mathfrak_2; e.g., Serre uses the fact that an \mathfrak_2-module with a primitive element of negative weight is infinite-dimensional, contradicting \dim \mathfrak g < \infty.) Let h_ \in \mathfrak, e_ \in \mathfrak_, f_ \in \mathfrak_ with the commutation relations _, f_= h_,
_, e_ The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= 2e_, _, f_= -2f_; i.e., the h_, e_, f_ correspond to the standard basis of \mathfrak_2. The linear functionals in \Phi are called the roots of \mathfrak g relative to \mathfrak h. The roots span \mathfrak h^* (since if \alpha(h) = 0, \alpha \in \Phi, then \operatorname(h) is the zero operator; i.e., h is in the center, which is zero.) Moreover, from the representation theory of \mathfrak_2, one deduces the following symmetry and integral properties of \Phi: for each \alpha, \beta \in \Phi, Note that s_ has the properties (1) s_(\alpha) = -\alpha and (2) the fixed-point set is \, which means that s_ is the reflection with respect to the hyperplane corresponding to \alpha. The above then says that \Phi is a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
. It follows from the general theory of a root system that \Phi contains a basis \alpha_1, \dots, \alpha_l of \mathfrak^* such that each root is a linear combination of \alpha_1, \dots, \alpha_l with integer coefficients of the same sign; the roots \alpha_i are called simple roots. Let e_i = e_, etc. Then the 3l elements e_i, f_i, h_i (called Chevalley generators) generate \mathfrak g as a Lie algebra. Moreover, they satisfy the relations (called Serre relations): with a_ = \alpha_j(h_i), : _i, h_j= 0, : _i, f_i= h_i, _i, f_j= 0, i \ne j, : _i, e_j= a_ e_j, _i, f_j= -a_ f_j, :\operatorname(e_i)^(e_j) = \operatorname(f_i)^(f_j) = 0, i \ne j. 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 _ 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 \mathfrak^* \simeq \mathfrak generated by the s_\alpha's. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of \mathfrak are invariant under the Weyl group.


Example root space decomposition in sln(C)

For \mathfrak = \mathfrak_n(\mathbb) and the Cartan subalgebra \mathfrak of diagonal matrices, define \lambda_i \in \mathfrak^* by :\lambda_i(d(a_1,\ldots, a_n)) = a_i, where d(a_1,\ldots, a_n) denotes the diagonal matrix with a_1,\ldots, a_n on the diagonal. Then the decomposition is given by :\mathfrak = \mathfrak\oplus \left( \bigoplus_ \mathfrak_ \right) where :\mathfrak_ = \text_\mathbb(e_) for the vector e_ in \mathfrak_n(\mathbb) with the standard (matrix) basis, meaning e_ represents the basis vector in the i-th row and j-th column. This decomposition of \mathfrak has an associated root system: :\Phi = \


sl2(C)

For example, in \mathfrak_2(\mathbb) the decomposition is :\mathfrak_2= \mathfrak\oplus \mathfrak_\oplus \mathfrak_ and the associated root system is :\Phi = \


sl3(C)

In \mathfrak_3(\mathbb) the decomposition is :\mathfrak_3 = \mathfrak \oplus \mathfrak_ \oplus \mathfrak_ \oplus \mathfrak_ \oplus \mathfrak_ \oplus \mathfrak_ \oplus \mathfrak_ and the associated root system is given by :\Phi = \


Examples

As noted in #Structure, semisimple
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s over \mathbb (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 diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s, are: * A_n: \mathfrak _, the special linear Lie algebra. * B_n: \mathfrak_, the odd-dimensional
special orthogonal Lie algebra In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. * C_n: \mathfrak _, the symplectic Lie algebra. * D_n: \mathfrak_, the even-dimensional
special orthogonal Lie algebra In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
(n>1). The restriction n>1 in the D_n family is needed because \mathfrak_ is one-dimensional and commutative and therefore not semisimple. These Lie algebras are numbered so that ''n'' is the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
. 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 \mathfrak_ \cong \mathfrak_ \oplus \mathfrak_ and \mathfrak_ \cong \mathfrak_. These four families, together with five exceptions ( E6, E7, E8, F4, and G2), 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 The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – An, Bn, Cn, and Dn – with five exceptions E6, E7, E8, F4, and G2. Simple Lie algebras are classified by the connected
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s, 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 In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
(see below) and its
adjoint action In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
on the Lie algebra. The
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
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 In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
, which is significantly more complicated. The enumeration of the four families is non-redundant and consists only of simple algebras if n \geq 1 for An, n \geq 2 for Bn, n \geq 3 for Cn, and n \geq 4 for Dn. If one starts numbering lower, the enumeration is redundant, and one has
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
s 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 diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
s, which are Dynkin diagrams with additional data ("decorations").


Representation theory of semisimple Lie algebras

Let \mathfrak g be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in #Structure, \mathfrak g = \mathfrak h \oplus \bigoplus_ \mathfrak g_ where \Phi is the root system. Choose the simple roots in \Phi; a root \alpha of \Phi is then called
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
and is denoted by \alpha > 0 if it is a linear combination of the simple roots with non-negative integer coefficients. Let \mathfrak b = \mathfrak h \oplus \bigoplus_ \mathfrak g_, which is a maximal solvable subalgebra of \mathfrak g, the
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
. Let ''V'' be a (possibly-infinite-dimensional) simple \mathfrak g-module. If ''V'' happens to admit a \mathfrak b-weight vector v_0, then it is unique up to scaling and is called the
highest weight vector In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...
of ''V''. It is also an \mathfrak h-weight vector and the \mathfrak h-weight of v_0, a linear functional of \mathfrak h, is called the highest weight of ''V''. The basic yet nontrivial facts then are (1) to each linear functional \mu \in \mathfrak h^*, there exists a simple \mathfrak g-module V^ having \mu as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between \mathfrak h^* and the set of the equivalence classes of simple \mathfrak g-modules admitting a Borel-weight vector. For applications, one is often interested in a finite-dimensional simple \mathfrak g-module (a finite-dimensional irreducible representation). This is especially the case when \mathfrak g is the Lie algebra of a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
(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 C \subset \mathfrak^*, we mean the convex cone C = \ where h_ \in mathfrak g_, \mathfrak g_/math> is a unique vector such that \alpha(h_) = 2. The criterion then reads: *\dim V^ < \infty if and only if, for each positive root \alpha > 0, (1) \mu(h_) is an integer and (2) \mu lies in C. A linear functional \mu 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 \mathfrak g-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 In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
. The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
of a semisimple Lie algebra \mathfrak g is completely reducible; i.e., it is a direct sum of simple \mathfrak g-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 \mathfrak g be a finite-dimensional real semisimple Lie algebra and \mathfrak^ = \mathfrak \otimes_ \mathbb the complexification of it (which is again semisimple). The real Lie algebra \mathfrak g is called a real form of \mathfrak^. 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 \mathfrak g is a compact form and \mathfrak h \subset \mathfrak g a maximal abelian subspace. One can show (for example, from the fact \mathfrak g is the Lie algebra of a compact Lie group) that \operatorname(\mathfrak h) consists of skew-Hermitian matrices, diagonalizable over \mathbb with imaginary eigenvalues. Hence, \mathfrak h^ is a
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
of \mathfrak^ and there results in the root space decomposition (cf. #Structure) :\mathfrak^ = \mathfrak^ \oplus \bigoplus_ \mathfrak_ where each \alpha \in \Phi is real-valued on i \mathfrak; thus, can be identified with a real-linear functional on the real vector space i \mathfrak. For example, let \mathfrak = \mathfrak(n) and take \mathfrak h \subset \mathfrak g the subspace of all diagonal matrices. Note \mathfrak^ = \mathfrak_n \mathbb. Let e_i be the linear functional on \mathfrak^ given by e_i(H) = h_i for H = \operatorname(h_1, \dots, h_n). Then for each H \in \mathfrak^, : , E_= (e_i(H) - e_j(H)) E_ where E_ is the matrix that has 1 on the (i, j)-th spot and zero elsewhere. Hence, each root \alpha is of the form \alpha = e_i - e_j, i \ne j and the root space decomposition is the decomposition of matrices: :\mathfrak^ = \mathfrak^ \oplus \bigoplus_ \mathbb E_.


Noncompact case

Suppose \mathfrak g 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 \theta and let \mathfrak g = \mathfrak k \oplus \mathfrak p be the eigenspace decomposition of \theta, where \mathfrak k, \mathfrak p are the eigenspaces for 1 and -1, respectively. For example, if \mathfrak g = \mathfrak_n \mathbb and \theta the negative transpose, then \mathfrak k = \mathfrak(n). Let \mathfrak a \subset \mathfrak p be a maximal abelian subspace. Now, \operatorname(\mathfrak p) consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in \operatorname(\mathfrak a) 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): :\mathfrak g = \mathfrak g_0 \oplus \bigoplus_ \mathfrak_ where *the elements in \Phi are called the restricted roots, *\theta(\mathfrak_) = \mathfrak_ for any linear functional \alpha; in particular, -\Phi \subset \Phi, *\mathfrak g_0 = \mathfrak a \oplus Z_(\mathfrak a). Moreover, \Phi is a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
but not necessarily reduced one (i.e., it can happen \alpha, 2\alpha are both roots).


The case of sl(n,C)

If \mathfrak=\mathrm(n,\mathbb), then \mathfrak may be taken to be the diagonal subalgebra of \mathfrak, consisting of diagonal matrices whose diagonal entries sum to zero. Since \mathfrak has dimension n-1, we see that \mathrm(n;\mathbb) has rank n-1. The root vectors X in this case may be taken to be the matrices E_ with i\neq j, where E_ is the matrix with a 1 in the (i,j) spot and zeros elsewhere. If H is a diagonal matrix with diagonal entries \lambda_1,\ldots,\lambda_n, then we have : ,E_(\lambda_i-\lambda_j)E_. Thus, the roots for \mathrm(n,\mathbb) are the linear functionals \alpha_ given by :\alpha_(H)=\lambda_i-\lambda_j. After identifying \mathfrak with its dual, the roots become the vectors \alpha_:=e_i-e_j in the space of n-tuples that sum to zero. This is the root system known as A_ in the conventional labeling. The reflection associated to the root \alpha_ acts on \mathfrak by transposing the i and j diagonal entries. The Weyl group is then just the permutation group on n elements, acting by permuting the diagonal entries of matrices in \mathfrak.


Generalizations

Semisimple Lie algebras admit certain generalizations. Firstly, many statements that are true for semisimple Lie algebras are true more generally for
reductive Lie algebra In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfrak ...
s. 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
; for example, \mathfrak_n is semisimple, and \mathfrak_n 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 semisimple Lie algebras over algebraically closed fields, for instance, the splitting Cartan subalgebra playing the same role as the
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
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 GL_n(\mathbb) of symmetries of an ''n''-dimensional real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(equivalently, the group of invertible matrices) is reductive.


See also

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Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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Root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
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Lie algebra representation In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket ...
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Compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
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Simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
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Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
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Jacobson–Morozov theorem In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , . Statement The statement of Jacobson–Morozov relies on t ...


References

* * . * * . * * *. * . {{DEFAULTSORT:Semisimple Lie Algebra Properties of Lie algebras