View source for Lie algebra, semi-simple

A Lie algebra that has no non-zero solvable ideals (see [[Lie algebra, solvable|Lie algebra, solvable]]). Henceforth finite-dimensional semi-simple Lie algebras over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585101.png" /> of characteristic 0 are considered (for semi-simple Lie algebras over a field of non-zero characteristic see [[Lie algebra|Lie algebra]]).

The fact that a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585102.png" /> is semi-simple is equivalent to any of the following conditions:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585103.png" /> does not contain non-zero Abelian ideals;

2) the [[Killing form|Killing form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585104.png" /> is non-singular (Cartan's criterion);

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585105.png" /> splits into the direct sum of non-Abelian simple ideals;

4) every finite-dimensional linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585106.png" /> is completely reducible (in other words: every finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585107.png" />-module is semi-simple);

5) the one-dimensional cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585108.png" /> with values in an arbitrary finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585109.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851010.png" /> be a semi-simple Lie algebra over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851011.png" />. The adjoint representation maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851012.png" /> isomorphically onto the linear Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851013.png" />, which is the Lie algebra of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851014.png" /> of all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851015.png" /> and is therefore an algebraic Lie algebra (cf. [[Lie algebra, algebraic|Lie algebra, algebraic]]). An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851016.png" /> is said to be semi-simple (nilpotent) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851017.png" /> is semi-simple (respectively, nilpotent). This property of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851018.png" /> is preserved by any homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851019.png" /> into another semi-simple Lie algebra. The identity component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851020.png" /> coincides with the group of inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851021.png" />, that is, it is generated by the automorphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851023.png" />.

In the study of semi-simple Lie algebras over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851024.png" /> an important role is played by the roots of a semi-simple Lie algebra, which are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851025.png" /> be a [[Cartan subalgebra|Cartan subalgebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851026.png" />. For a non-zero linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851027.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851028.png" /> denote the linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851029.png" /> given by the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851032.png" /> is called a root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851033.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851034.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851035.png" /> of all non-zero roots is called the root system, or system of roots, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851036.png" />. One has the root decomposition

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The root system and the root decomposition of a semi-simple Lie algebra have the following properties:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851038.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851039.png" /> and is a reduced [[Root system|root system]] in the abstract sense (in the linear hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851040.png" /> over the field of the real numbers). The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851041.png" /> is irreducible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851042.png" /> is simple.

b) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851043.png" />,

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There is a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851046.png" />.

c) For every non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851047.png" /> there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851049.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png" /></td> </tr></table>

Moreover,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851052.png" /> is the scalar product induced by the Killing form.

d) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851056.png" /> are orthogonal with respect to the Killing form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png" />.

A basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851058.png" /> of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851059.png" /> is also called a system of simple roots of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851060.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851061.png" /> be the system of positive roots with respect to the given basis and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851062.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851063.png" />. Then the elements

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form a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851065.png" />, called a Cartan basis. On the other hand, the elements

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form a system of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851067.png" />, and the defining relations have the following form:

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851071.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851072.png" /> and

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Property d) implies that

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851075.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851076.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851077.png" /> can be chosen in such a way that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851079.png" /> is the largest integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851080.png" />. The corresponding Cartan basis is called a Chevalley basis. The structure constants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851081.png" /> in this basis are integers, which makes it possible to associate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851082.png" /> Lie algebras and algebraic groups (see [[Chevalley group|Chevalley group]]) over fields of arbitrary characteristic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851083.png" />, then the linear hull over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851084.png" /> of the vectors

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png" /></td> </tr></table>

is a compact real form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851086.png" />.

A semi-simple Lie algebra is defined up to an isomorphism by its Cartan subalgebra and the corresponding root system. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851088.png" /> are semi-simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851091.png" /> are their Cartan subalgebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851093.png" /> are the corresponding root systems, then every isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851094.png" /> that induces an isomorphism of the root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851096.png" /> can be extended to an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851097.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851098.png" /> essentially coincides with the classification of reduced root systems (respectively, irreducible reduced root systems).

Simple Lie algebras that correspond to root systems of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851099.png" />–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510100.png" /> are said to be classical and have the following form.

Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510102.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510103.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510104.png" /> with trace 0; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510105.png" />.

Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510107.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510108.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510109.png" /> that are skew-symmetric with respect to a given non-singular symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510110.png" />.

Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510112.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510113.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510114.png" /> that are skew-symmetric with respect to a given non-singular skew-symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510115.png" />.

Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510117.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510118.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510119.png" /> that are skew-symmetric with respect to a given non-singular symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510120.png" />.

The simple Lie algebras corresponding to the root systems of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510125.png" /> are called special, or exceptional (see [[Lie algebra, exceptional|Lie algebra, exceptional]]).

The [[Cartan matrix|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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510126.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510128.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png" /></td> </tr></table>

The classification of split semi-simple Lie algebras over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510134.png" /> of characteristic zero (a semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510135.png" /> is said to be split if it has a Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510136.png" /> such that all characteristic roots of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510138.png" />, lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510139.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510140.png" />–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510141.png" /> have the form stated above, except that in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510143.png" /> one must consider non-singular symmetric bilinear forms with Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510144.png" />.

The problem of classifying arbitrary semi-simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510145.png" /> reduces to the following problem: To list, up to an isomorphism, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510146.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510147.png" />, that is, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510148.png" />-subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510149.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510150.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510151.png" /> is an algebraically closed extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510153.png" /> is a given semi-simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510154.png" />. The solution of this problem can also be obtained in terms of root systems (see [[Form of an algebraic group|Form of an algebraic group]]; [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510155.png" /> is a classical simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510156.png" /> (other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510157.png" />), there is another method of classifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510158.png" />-forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510159.png" />, based on an examination of simple associative algebras (see [[#References|[3]]]).

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510160.png" /> the classification of semi-simple Lie algebras goes as follows (see [[#References|[6]]], [[#References|[7]]]). Every simple non-Abelian Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510161.png" /> is either a simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510162.png" /> (regarded as an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510163.png" />), or the real form of a simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510164.png" />. The classification of real forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510165.png" /> in a simple classical Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510166.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510167.png" /> is as follows:

I) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510168.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510170.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510171.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510172.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510173.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510174.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510175.png" />, the subalgebra of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510176.png" /> that preserve a certain quaternion structure. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510177.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510178.png" />, the subalgebra of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510179.png" /> that are skew-symmetric with respect to a non-singular Hermitian form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510181.png" />

II) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510182.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510183.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510184.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510185.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510186.png" />, the algebra of a linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510187.png" /> that are skew-symmetric with respect to a non-singular symmetric bilinear form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510189.png" />.

III) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510190.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510192.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510193.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510194.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510195.png" /> that are skew-symmetric with respect to a non-singular skew-symmetric bilinear form. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510196.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510198.png" />, the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510199.png" /> consisting of transformations that preserve a certain quaternion structure.

IV) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510200.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510202.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510203.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510204.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510205.png" /> that are skew-symmetric with respect to a non-singular bilinear symmetric form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510207.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510208.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510209.png" />, the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510210.png" /> consisting of transformations that preserve a certain quaternion structure.

Semi-simple Lie algebras over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510211.png" /> 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 [[#References|[2]]]. In the papers of Killing and Cartan the roots of a Lie algebra appeared as the characteristic roots of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510212.png" />. 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|Globally symmetric Riemannian space]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I"  ''Math. Ann.'' , '''31'''  (1888)  pp. 252–290</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II"  ''Math. Ann.'' , '''33'''  (1889)  pp. 1–48</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III"  ''Math. Ann.'' , '''34'''  (1889)  pp. 57–122</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV"  ''Math. Ann.'' , '''36'''  (1890)  pp. 161–189</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Cartan,   "Sur la structure des groupes de transformations finis et continues" , ''Oeuvres complètes'' , '''1''' , Gauthier-Villars  (1952)  pp. 137–287</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.G. Steinberg,   "Lectures on Chevalley groups" , Yale Univ. Press  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S. Araki,   "On root systems and an infinitesimal classification of irreducible symmetric spaces"  ''Osaka J. Math.'' , '''13'''  (1962)  pp. 1–34</TD></TR></table>



====Comments====
The defining relations, mentioned above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510213.png" />, are known as the Serre relations.

It is customary to encode the information contained in the Cartan matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510214.png" />–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510215.png" /> by means of the so-called Dynkin diagrams.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510216.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510217.png" /> nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510218.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510219.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510220.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510221.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510222.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510223.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510224.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510225.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510226.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510227.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(6 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510228.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(7 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510229.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(8 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510230.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(4 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510231.png" /></td> <td colname="2" style="background-color:white;" colspan="1">

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

</td> <td colname="3" style="background-color:white;" colspan="1">(2 nodes)</td> </tr> </tbody> </table>

</td></tr> </table>

The rules for recovering the Cartan matrix from the corresponding Dynkin diagram (also called Dynkin graph occasionally) are as follows. Number the vertices, e.g.,

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l058510a.gif" />

Figure: l058510a

On the diagonal of the Cartan matrix all elements are equal to 2. If nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510233.png" /> are not directly linked, then the matrix entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510234.png" />. If two nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510235.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510236.png" /> are directly linked by a single edge, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510237.png" />. If two nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510238.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510239.png" /> are directly linked by a double, respectively triple, edge and the arrow points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510240.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510241.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510242.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510243.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510244.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510245.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.B. Frenkel,   V.G. Kac,   "Basic representations of affine Lie algebras and dual resonance models"  ''Invent. Math.'' , '''62'''  (1980)  pp. 23–66</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Serre,   "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR></table>