# Lie algebra, semi-simple

A Lie algebra that has no non-zero solvable ideals (see Lie algebra, solvable). Henceforth finite-dimensional semi-simple Lie algebras over a field $k$ of characteristic 0 are considered (for semi-simple Lie algebras over a field of non-zero characteristic see Lie algebra).

The fact that a finite-dimensional Lie algebra $\mathfrak g$ is semi-simple is equivalent to any of the following conditions:

1) $\mathfrak g$ does not contain non-zero Abelian ideals;

2) the Killing form of $\mathfrak g$ is non-singular (Cartan's criterion);

3) $\mathfrak g$ splits into the direct sum of non-Abelian simple ideals;

4) every finite-dimensional linear representation of $\mathfrak g$ is completely reducible (in other words: every finite-dimensional $\mathfrak g$ -module is semi-simple);

5) the one-dimensional cohomology of $\mathfrak g$ with values in an arbitrary finite-dimensional $\mathfrak g$ -module is trivial.

Any ideal and any quotient algebra of a semi-simple Lie algebra is also semi-simple. The decomposition of a semi-simple Lie algebra mentioned in condition 3) is unique. A special case of condition 5) is the following assertion: All derivations of a semi-simple Lie algebra are inner. The property of a Lie algebra of being semi-simple is preserved by both extensions and restrictions of the ground field.

Let $\mathfrak g$ be a semi-simple Lie algebra over an algebraically closed field $k$ . The adjoint representation maps $\mathfrak g$ isomorphically onto the linear Lie algebra $\mathop{\rm ad}\nolimits \ \mathfrak g$ , which is the Lie algebra of the algebraic group $\mathop{\rm Aut}\nolimits \ \mathfrak g$ of all automorphisms of $\mathfrak g$ and is therefore an algebraic Lie algebra (cf. Lie algebra, algebraic). An element $X \in \mathfrak g$ is said to be semi-simple (nilpotent) if $\mathop{\rm ad}\nolimits \ X$ is semi-simple (respectively, nilpotent). This property of an element $X$ is preserved by any homomorphism of $\mathfrak g$ into another semi-simple Lie algebra. The identity component $( \mathop{\rm Aut}\nolimits \ \mathfrak g ) ^{0}$ coincides with the group of inner automorphisms of $\mathfrak g$ , that is, it is generated by the automorphisms of the form $\mathop{\rm exp}\nolimits ( \mathop{\rm ad}\nolimits \ X )$ , $X \in \mathfrak g$ .

In the study of semi-simple Lie algebras over an algebraically closed field $k$ an important role is played by the roots of a semi-simple Lie algebra, which are defined as follows. Let $\mathfrak h$ be a Cartan subalgebra of $\mathfrak g$ . For a non-zero linear function $\alpha \in \mathfrak h ^ \ast$ , let $\mathfrak g _ \alpha$ denote the linear subspace of $\mathfrak g$ given by the condition $$\mathfrak g _ \alpha = \{ {X \in \mathfrak g} : {[ H ,\ X ] = \alpha (H) X , H \in \mathfrak h} \} .$$ If $\mathfrak g _ \alpha \neq 0$ , then $\alpha$ is called a root of $\mathfrak g$ with respect to $\mathfrak h$ . The set $\Sigma$ of all non-zero roots is called the root system, or system of roots, of $\mathfrak g$ . One has the root decomposition $$\mathfrak g = \mathfrak h + \sum _ {\alpha \in \Sigma} \mathfrak g _ \alpha .$$ The root system and the root decomposition of a semi-simple Lie algebra have the following properties:

a) $\Sigma$ generates $\mathfrak h ^ \ast$ and is a reduced root system in the abstract sense (in the linear hull of $\Sigma$ over the field of the real numbers). The system $\Sigma$ is irreducible if and only if $\mathfrak g$ is simple.

b) For any $\alpha \in \Sigma$ , $$\mathop{\rm dim}\nolimits \ \mathfrak g _ \alpha = \mathop{\rm dim}\nolimits \ [ \mathfrak g _ \alpha ,\ \mathfrak g _ {- \alpha} ] = 1 .$$ There is a unique element $H _ \alpha \in [ \mathfrak g _ \alpha ,\ \mathfrak g _ {- \alpha} ]$ such that $\alpha ( H _ \alpha ) = 2$ .

c) For every non-zero $X _ \alpha \in \mathfrak g _ \alpha$ there is a unique $Y _ \alpha \in \mathfrak g _ {- \alpha}$ such that $[ X _ \alpha ,\ Y _ \alpha ] = H _ \alpha$ , and $$[ H _ \alpha ,\ X _ \alpha ] = 2 X _ \alpha \textrm{ and } [ H _ \alpha ,\ Y _ \alpha ] = - 2 Y _ \alpha .$$ Moreover, $$\beta ( H _ \alpha ) = \frac{2 ( \alpha ,\ \beta )}{( \alpha ,\ \alpha )} , \alpha ,\ \beta \in \Sigma ,$$ where $( \ ,\ )$ is the scalar product induced by the Killing form.

d) If $\alpha ,\ \beta \in \Sigma$ and $\alpha + \beta \neq 0$ , then $\mathfrak g _ \alpha$ and $\mathfrak g _ \beta$ are orthogonal with respect to the Killing form and $[ \mathfrak g _ \alpha ,\ \mathfrak g _ \beta ] = \mathfrak g _ {\alpha + \beta}$ .

A basis $\{ \alpha _{1} \dots \alpha _{n} \}$ of the root system $\Sigma$ is also called a system of simple roots of the algebra $\mathfrak g$ . Let $\Sigma _{+}$ be the system of positive roots with respect to the given basis and let $X _ {- \alpha} = Y _ \alpha$ $( \alpha \in \Sigma _{+} )$ . Then the elements $$H _ {\alpha _{1}} \dots H _ {\alpha _{k}} ,\ X _ \alpha ( \alpha \in \Sigma )$$ form a basis of $\mathfrak g$ , called a Cartan basis. On the other hand, the elements $$X _ {\alpha _{i}} , X _ {- \alpha _{i}} ( i = 1 \dots n )$$ form a system of generators of $\mathfrak g$ , and the defining relations have the following form: $$[ [ X _ {\alpha _{i}} ,\ X _ {- \alpha _{i}} ] ,\ X _ {\alpha _{j}} ] = n ( i ,\ j ) X _ {\alpha _{j}} ,$$ $$[ [ X _ {\alpha _{i}} ,\ X _ {- \alpha _{i}} ] ,\ X _ {- \alpha _{j}} ] = - n ( i ,\ j ) X _ {\alpha _{j}} ,$$ $$( \mathop{\rm ad}\nolimits \ X _ {\alpha _{i}} ) ^ {1 - n ( i ,\ j )} X _ {\alpha _{j}} = 0 ,$$ $$( \mathop{\rm ad}\nolimits \ X _ {- \alpha _{i}} ) ^ {1 - n ( i ,\ j )} X _ {- \alpha _{j}} = 0 .$$ Here $i ,\ j = 1 \dots n$ and $$n ( i ,\ j ) = \alpha _{j} ( H _{i} ) = \frac{2 ( \alpha _{i} ,\ \alpha _{j} )}{( \alpha _{j} ,\ \alpha _{j} )} .$$ Property d) implies that $$[ X _ \alpha ,\ X _ \beta ] = \left \{ \begin{array}{ll} N _ {\alpha , \beta} X _ {\alpha + \beta} & \textrm{ if } \alpha + \beta \in \Sigma , \\ 0 & \textrm{ if } \alpha + \beta \notin \Sigma , \\ \end{array} \right .$$where $N _ {\alpha , \beta} \in k$ . The elements $X _ \alpha$ $( \alpha \in \Sigma _{+} )$ can be chosen in such a way that $$N _ {\alpha , \beta} = - N _ {- \alpha , - \beta} \textrm{ and } N _ {\alpha , \beta} = \pm ( p + 1 ) ,$$ where $p$ is the largest integer such that $\beta - p \alpha \in \Sigma$ . The corresponding Cartan basis is called a Chevalley basis. The structure constants of $\mathfrak g$ in this basis are integers, which makes it possible to associate with $\mathfrak g$ Lie algebras and algebraic groups (see Chevalley group) over fields of arbitrary characteristic. If $k = \mathbf C$ , then the linear hull over $\mathbf R$ of the vectors $$i H _ \alpha , X _ \alpha - X _ {- \alpha} , i ( X _ \alpha + X _ {- \alpha} ) \quad ( \alpha \in \Sigma _{+} )$$ is a compact real form of $\mathfrak g$ .

A semi-simple Lie algebra is defined up to an isomorphism by its Cartan subalgebra and the corresponding root system. More precisely, if $\mathfrak g _{1}$ and $\mathfrak g _{2}$ are semi-simple Lie algebras over $k$ , $\mathfrak h _{1}$ and $\mathfrak h _{2}$ are their Cartan subalgebras and $\Sigma _{1}$ and $\Sigma _{2}$ are the corresponding root systems, then every isomorphism $\mathfrak h _{1} \rightarrow \mathfrak h _{2}$ that induces an isomorphism of the root systems $\Sigma _{1}$ and $\Sigma _{2}$ can be extended to an isomorphism $\mathfrak g _{1} \rightarrow \mathfrak g _{2}$ . On the other hand, any reduced root system can be realized as the root system of some semi-simple Lie algebra. Thus, the classification of semi-simple Lie algebras (respectively, simple non-Abelian Lie algebras) over an algebraically closed field $k$ essentially coincides with the classification of reduced root systems (respectively, irreducible reduced root systems).

Simple Lie algebras that correspond to root systems of types $A$ – $D$ are said to be classical and have the following form.

Type $A _{n}$ , $n \geq 1$ . $\mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\ k )$ , the algebra of linear transformations of the space $k ^{n+1}$ with trace 0; $\mathop{\rm dim}\nolimits \ \mathfrak g = n ( n + 2 )$ .

Type $B _{n}$ , $n \geq 2$ . $\mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\ k )$ , the algebra of linear transformations of the space $k ^{2n+1}$ that are skew-symmetric with respect to a given non-singular symmetric bilinear form; $\mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n + 1 )$ .

Type $C _{n}$ , $n \geq 3$ . $\mathfrak g = \mathfrak s \mathfrak p ( n ,\ k )$ , the algebra of linear transformations of the space $k ^{2n}$ that are skew-symmetric with respect to a given non-singular skew-symmetric bilinear form; $\mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n + 1 )$ .

Type $D _{n}$ , $n \geq 4$ . $\mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\ k )$ , the algebra of linear transformations of the space $k ^{2n}$ that are skew-symmetric with respect to a given non-singular symmetric bilinear form; $\mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n - 1 )$ .

The simple Lie algebras corresponding to the root systems of types $E _{6}$ , $E _{7}$ , $E _{8}$ , $F _{4}$ , $G _{2}$ are called special, or exceptional (see Lie algebra, exceptional).

The Cartan matrix of a semi-simple Lie algebra over an algebraically closed field also determines this algebra uniquely up to an isomorphism. The Cartan matrices of the simple Lie algebras have the following form: $$A _{n} : \left \| \begin{array}{rrrcr} 2 &-1 & 0 &\dots & 0 \\ -1 & 2 &-1 &\dots & 0 \\ 0 &-1 & 2 &\dots & 0 \\ . & . & . &\ddots & . \\ 0 & 0 & 0 &\dots &-1 \\ 0 & 0 & 0 &\dots & 2 \\ \end{array} \right \| ,$$ $$B _{n} : \left \| \begin{array}{rrrrrr} 2 &-1 & 0 &\dots & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 \\ . & . & . &\ddots & . & . \\ 0 & 0 & 0 &\dots & 2 &-2 \\ 0 & 0 & 0 &\dots &-1 & 2 \\ \end{array} \right \| ,$$ $$C _{n} : \left \| \begin{array}{rrrrrrr} 2 &-1 & 0 &\dots & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 \\ . & . & . &\cdots & . & . \\ 0 & 0 & 0 &\dots & 2 &-1 \\ 0 & 0 & 0 &\dots &-2 & 2 \\ \end{array} \right \| ,$$ $$D _{n} : \left \| \begin{array}{rrrrrrrr} 2 &-1 & 0 &\dots & 0 & 0 & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 & 0 & 0 \\ . & . & . &\ddots & . & . & . & . \\ 0 & 0 & 0 &\dots & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 &\dots &-1 & 2 &-1 &-1 \\ 0 & 0 & 0 &\dots & 0 &-1 & 2 & 0 \\ 0 & 0 & 0 &\dots & 0 &-1 & 0 & 2 \\ \end{array} \right \| ,$$ $$E _{6} : \left \| \begin{array}{rrrrrr} 2 & 0 &-1 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| ,$$ $$E _{7} : \left \| \begin{array}{rrrrrrr} 2 & 0 &-1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| ,$$ $$E _{8} : \left \| \begin{array}{rrrrrrrr} 2 & 0 &-1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 & 0 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 & 0 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| ,$$ $$F _{4} : \left \| \begin{array}{rrrr} 2 &-1 & 0 & 0 \\ -1 & 2 &-2 & 0 \\ 0 &-1 & 2 &-1 \\ 0 & 0 &-1 & 2 \\ \end{array} \right \| , G _{2} : \left \| \begin{array}{rr} 2 &-1 \\ -3 & 2 \\ \end{array} \right \| .$$ The classification of split semi-simple Lie algebras over an arbitrary field $k$ of characteristic zero (a semi-simple Lie algebra $\mathfrak g$ is said to be split if it has a Cartan subalgebra $\mathfrak h \subset \mathfrak g$ such that all characteristic roots of the operators $\mathop{\rm ad}\nolimits \ X$ , $X \in \mathfrak h$ , lie in $k$ ) goes in the same way as in the case of an algebraically closed field. Namely, to every irreducible reduced root system corresponds a unique split semi-simple Lie algebra. In particular, split semi-simple Lie algebras of types $A$ – $D$ have the form stated above, except that in the cases $B$ and $D$ one must consider non-singular symmetric bilinear forms with Witt index $n$ .

The problem of classifying arbitrary semi-simple Lie algebras over $k$ reduces to the following problem: To list, up to an isomorphism, all $k$ -forms $\mathfrak g _{0} \subset \mathfrak g$ , that is, all $k$ -subalgebras $\mathfrak g _{0} \subset \mathfrak g$ such that $\mathfrak g = \mathfrak g _{0} \otimes _{k} K$ . Here $K$ is an algebraically closed extension of $k$ and $\mathfrak g$ is a given semi-simple Lie algebra over $K$ . The solution of this problem can also be obtained in terms of root systems (see Form of an algebraic group; Form of an (algebraic) structure). When $\mathfrak g$ is a classical simple Lie algebra over $k$ (other than $D _{4}$ ), there is another method of classifying $k$ -forms in $\mathfrak g$ , based on an examination of simple associative algebras (see ).

When $k = \mathbf R$ the classification of semi-simple Lie algebras goes as follows (see , ). Every simple non-Abelian Lie algebra over $\mathbf R$ is either a simple Lie algebra over $\mathbf C$ (regarded as an algebra over $\mathbf R$ ), or the real form of a simple Lie algebra over $\mathbf C$ . The classification of real forms $\mathfrak g _{0}$ in a simple classical Lie algebra $\mathfrak g$ over $\mathbf C$ is as follows:

I) Type $A _{n}$ : $\mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\ \mathbf C )$ , $n \geq 1$ . $A _{I}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak l ( n + 1 ,\ \mathbf R )$ . $A _{II}$ : $n + 1 = 2 m$ is even, $\mathfrak g _{0} = \mathfrak s \mathfrak u ^ \ast ( 2 n )$ , the subalgebra of elements of $\mathfrak s \mathfrak l ( 2 m ,\ \mathbf C )$ that preserve a certain quaternion structure. $A _{III}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak u ( p ,\ n + 1 - p )$ , the subalgebra of elements of $\mathfrak s \mathfrak l ( n + 1 ,\ \mathbf C )$ that are skew-symmetric with respect to a non-singular Hermitian form of positive index $p$ , $0 \leq p \leq ( n + 1 ) / 2 .$

II) Type $B _{n}$ : $\mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\ \mathbf C )$ , $n \geq 2$ . $B _{I}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\ 2 n + 1 - p )$ , the algebra of a linear transformations of the space $\mathbf R ^{2n+1}$ that are skew-symmetric with respect to a non-singular symmetric bilinear form of positive index $p$ , $0 \leq p \leq n$ .

III) Type $C _{n}$ : $\mathfrak g = \mathfrak s \mathfrak p (n,\ \mathbf C )$ , $n \geq 3$ . $C _{I}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak p ( n ,\ \mathbf R )$ , the algebra of linear transformations of the space $\mathbf R ^{2n}$ that are skew-symmetric with respect to a non-singular skew-symmetric bilinear form. $C _{II}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak p ( p ,\ n - p )$ , $0 \leq p \leq n / 2$ , the subalgebra of $\mathfrak s \mathfrak u ( 2 p ,\ 2 ( n - p ) )$ consisting of transformations that preserve a certain quaternion structure.

IV) Type $D _{n}$ : $\mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\ \mathbf C )$ , $n \geq 4$ . $D _{I}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\ 2 n - p )$ , the algebra of linear transformations of the space $\mathbf R ^{2n}$ that are skew-symmetric with respect to a non-singular bilinear symmetric form of positive index $p$ , $0 \leq p \leq n$ . $D _{III}$ : $\mathfrak g _{0} = \mathfrak s \mathfrak o ^ \ast ( 2 n ,\ \mathbf C )$ , the subalgebra of $\mathfrak s \mathfrak o (2n ,\ \mathbf C )$ consisting of transformations that preserve a certain quaternion structure.

Semi-simple Lie algebras over the field $\mathbf C$ were first considered in papers by W. Killing , who gave a classification of them, although in his proofs there were gaps, which were filled by E. Cartan . In the papers of Killing and Cartan the roots of a Lie algebra appeared as the characteristic roots of the operator $\mathop{\rm ad}\nolimits \ X$ . Cartan also gave a classification of real semi-simple Lie algebras by establishing a deep connection between these algebras and globally symmetric Riemannian spaces (cf. Globally symmetric Riemannian space).

How to Cite This Entry:
Lie algebra, semi-simple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_semi-simple&oldid=53489
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article