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A connected [[Lie group|Lie group]] that does not contain non-trivial connected solvable (or, equivalently, connected Abelian) normal subgroups. A connected Lie group is semi-simple if and only if its Lie algebra is semi-simple (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]). A connected Lie group $k$ is said to be simple if its Lie algebra is simple, that is, if $k$ does not contain non-trivial connected normal subgroups other than $k$. A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian normal subgroups.

The classification of semi-simple Lie groups reduces to the local classification, that is, to the classification of semi-simple Lie algebras (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), and also to the global classification of the Lie groups $k$ that correspond to a given semi-simple Lie algebra $8$.

In the case of Lie groups over the field $m$ of complex numbers the main result of the local classification is that every simply-connected simple non-Abelian complex Lie group is isomorphic to one of the groups $SL _ { \mathscr { K } } + 1$, $n \geq 1$, $( C )$, $n \geq 5$ (the universal covering of the group $n ( C )$), $p _ { Y } ( C )$, $n \geq 3$ (see [[Classical group|Classical group]]), or one of the exceptional complex Lie groups (see [[Lie algebra, exceptional|Lie algebra, exceptional]]). The global classification of the Lie groups corresponding to a semi-simple Lie algebra $8$ over $m$ goes as follows. Let $h$ be a [[Cartan subalgebra|Cartan subalgebra]] of $8$ and let $2$ be the [[Root system|root system]] of $8$ with respect to $h$. To every semi-simple Lie group $k$ with Lie algebra $8$ corresponds a lattice $\Gamma ( G ) \subset \mathfrak { h }$ that is the kernel of the [[Exponential mapping|exponential mapping]] $\operatorname { exp } : \mathfrak { h } \rightarrow G$. In particular, if $k$ is simply connected, then $\Gamma ( G )$ coincides with the lattice $\Gamma _ { 0 } = \Gamma _ { 0 } ( \mathfrak { g } )$ generated by the elements $2 \pi i H _ { \alpha }$, $\alpha \in \Sigma$ (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), and if $k$ is a group without centre (an adjoint group), then $\Gamma ( G )$ is the lattice

\begin{equation} \Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \} \end{equation}

In the general case $\Gamma _ { 0 } \subset \Gamma ( G ) \subset \Gamma _ { 1 }$. For any additive subgroup $M \subset b$ satisfying the condition $\Gamma _ { 0 } \subset M \subset \Gamma _ { 1 }$ there is a unique (up to isomorphism) connected Lie group $k$ with Lie algebra $8$ such that $\Gamma ( G ) = M$. The centre of $k$ is isomorphic to $\Gamma _ { 1 } / \Gamma ( G )$, and for the [[Fundamental group|fundamental group]] one has:

\begin{equation} \pi _ { 1 } ( G ) \cong \Gamma ( G ) / \Gamma _ { 0 } \end{equation}

The quotient group $Z _ { g } = \Gamma _ { 1 } / \Gamma _ { 0 }$ (the centre of the simply-connected Lie group with Lie algebra $8$) is finite and for the different types of simple Lie algebras $8$ it has the following form:

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellpadding="4" cellspacing="1" style="background-color:black;">  <tr> <td colname="1" colspan="1" style="background-color:white;">$8$</td> <td colname="2" colspan="1" style="background-color:white;">$A _ { n }$</td> <td colname="3" colspan="1" style="background-color:white;">$B _ { y }$</td> <td colname="4" colspan="1" style="background-color:white;">$C$</td> <td colname="5" colspan="1" style="background-color:white;">$D _ { 2 }$</td> <td colname="6" colspan="1" style="background-color:white;">$D _ { 2 } n + 1$</td> <td colname="7" colspan="1" style="background-color:white;">$E _ { 0 }$</td> <td colname="8" colspan="1" style="background-color:white;">$E _ { 7 }$</td> <td colname="9" colspan="1" style="background-color:white;">$E _ { 8 }$, $F _ { 4 }$, $G$</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">$Z _ { D }$</td> <td colname="2" colspan="1" style="background-color:white;">$z _ { n + 1 }$</td> <td colname="3" colspan="1" style="background-color:white;">$22$</td> <td colname="4" colspan="1" style="background-color:white;">$22$</td> <td colname="5" colspan="1" style="background-color:white;">$Z _ { 2 } \oplus Z _ { 2 }$</td> <td colname="6" colspan="1" style="background-color:white;">$24$</td> <td colname="7" colspan="1" style="background-color:white;">$23$</td> <td colname="8" colspan="1" style="background-color:white;">$22$</td> <td colname="9" colspan="1" style="background-color:white;">$0$</td> </tr>  </table>
</td></tr> </table>

The order of the group $\Gamma _ { 1 } / \Gamma _ { 0 }$ is the same as the number of vertices with coefficient 1 in the extended [[Dynkin diagram|Dynkin diagram]] of $8$; discarding one of the vertices gives the Dynkin diagram. A similar classification holds for compact real semi-simple Lie groups, each of which is imbedded in a unique complex semi-simple Lie group as a maximal compact subgroup (see [[Lie group, compact|Lie group, compact]]).

The global classification of non-compact real semi-simple Lie groups can be carried out in a similar but more complicated way. In particular, the centre $Z _ { D }$ of the simply-connected Lie group corresponding to a semi-simple Lie algebra $8$ over $R$ can be calculated as follows. Let $\mathfrak { g } = \mathfrak { k } + \mathfrak { p }$ be the [[Cartan decomposition|Cartan decomposition]], where $3$ is a maximal compact subalgebra of $8$ and $t$ is its orthogonal complement with respect to the Killing form, let $6$ be the corresponding involutive automorphism, extended to $g ^ { C }$, $h$ the Cartan subalgebra of $g ^ { C }$ containing a Cartan subalgebra $h ^ { \prime } \subset k$, $\theta _ { 0 }$ an automorphism of $g ^ { C }$ that coincides with $6$ on the roots with respect to $h$ and extended to the root vectors in an appropriate way, and $\mathfrak { g } 0 = \mathfrak { k } _ { 0 } + \mathfrak { p } _ { 0 }$ the Cartan decomposition of the real form $\mathfrak { g } \subset \mathfrak { g } ^ { \mathfrak { C } }$ corresponding to $\theta _ { 0 }$. Then $Z _ { g } \cong \Gamma _ { 1 } ( f _ { 0 } ) / \Gamma _ { 0 } [ e , t ]$ (see [[#References|[3]]], where this group is calculated for all types of simple algebras $8$ over $R$).

Every complex semi-simple Lie group $k$ has the unique structure of an affine algebraic group compatible with the analytic structure specified on it, and any analytic homomorphism of $k$ to an algebraic group is rational. The corresponding algebra of regular functions on $k$ coincides with the algebra of holomorphic representation functions. On the other hand, a non-compact real semi-simple Lie group does not always admit a faithful linear representation — the simplest example is the simply-connected Lie group corresponding to the Lie algebra $s l ( 2 , R )$. If $8$ is a semi-simple Lie algebra over $R$, then in the centre $Z _ { D }$ of the simply-connected group $G$ corresponding to $8$ there is a smallest subgroup $L ( \mathfrak { g } )$, called the linearizer, such that $G _ { 0 } / L ( \mathfrak { g } )$ is isomorphic to a linear semi-simple Lie group. If $u = \mathfrak { l } + \dot { \mathfrak { i } } \mathfrak { u }$ is the compact real form of $g ^ { C }$, then

\begin{equation} L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] ) \end{equation}

(see [[#References|[3]]], where this group is calculated for all types of simple Lie algebras $8$).

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> J.F. Adams, "Lectures on Lie groups" , Benjamin (1969) {{MR|0252560}} {{ZBL|0206.31604}} </td></tr><tr><td valign="top">[2]</td> <td valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </td></tr><tr><td valign="top">[3]</td> <td valign="top"> A.I. Sirota, A.S. Solodovnikov, "Noncompact semisimple Lie groups" ''Russian Math. Surveys'' , '''18''' : 3 (1963) pp. 85–140 ''Uspekhi Mat. Nauk'' , '''18''' : 3 (1963) pp. 87–144 {{MR|155929}} {{ZBL|0132.02101}} </td></tr></table>



====Comments====
The (infinite-dimensional) representation theory of semi-simple Lie groups over $R$ has been created for a large part by Harish-Chandra. See also the excellent survey of Harish-Chandra's work by V.S. Varadarajan in the collected works [[#References|[a1]]].

The $p$ reductions of semi-simple Lie groups over $2$ are called the Chevalley groups (cf. [[Chevalley group|Chevalley group]]), and from them most of the finite simple groups can be obtained (with the exception of the [[Alternating group|alternating group]] and the 26 sporadic groups, cf. [[Sporadic simple group|Sporadic simple group]]). For a survey of the structure and representation theory of the Chevalley groups see [[#References|[a2]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> Harish-Chandra, "Collected works" , '''1–4''' , Springer (1984) {{MR|}} {{ZBL|0699.62084}} {{ZBL|0653.01018}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> R.W. Carter, "Finite groups of Lie type: Conjugacy classes and complex characters" , Wiley (Interscience) (1985)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1974) {{MR|0343214}} {{ZBL|0281.53034}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) {{MR|0207883}} {{ZBL|0131.02702}} </td></tr></table>

[[Category:Lie theory and generalizations]]