# User:Maximilian Janisch/latexlist/Algebraic Groups/Lie group, semi-simple

A connected 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). 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), 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), or one of the exceptional complex Lie groups (see 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 of $8$ and let $2$ be the 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 $\operatorname { exp } : \mathfrak { h } \rightarrow G$. 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), 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 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:

 $8$ $A _ { n }$ $B _ { y }$ $C$ $D _ { 2 }$ $D _ { 2 } n + 1$ $E _ { 0 }$ $E _ { 7 }$ $E _ { 8 }$, $F _ { 4 }$, $G$ $Z _ { D }$ $z _ { n + 1 }$ $22$ $22$ $Z _ { 2 } \oplus Z _ { 2 }$ $24$ $23$ $22$ $0$

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 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).

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, 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 , 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 , where this group is calculated for all types of simple Lie algebras $8$).

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Maximilian Janisch/latexlist/Algebraic Groups/Lie group, semi-simple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/latexlist/Algebraic_Groups/Lie_group,_semi-simple&oldid=44031