View source for Lie group, semi-simple

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586801.png" /> is said to be simple if its Lie algebra is simple, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586802.png" /> does not contain non-trivial connected normal subgroups other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586803.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586804.png" /> that correspond to a given semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586805.png" />.

In the case of Lie groups over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586806.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l0586809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868010.png" /> (the universal covering of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868011.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868013.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868015.png" /> goes as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868016.png" /> be a [[Cartan subalgebra|Cartan subalgebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868017.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868018.png" /> be the [[Root system|root system]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868020.png" />. To every semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868021.png" /> with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868022.png" /> corresponds a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868023.png" /> that is the kernel of the [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868024.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868025.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868026.png" /> coincides with the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868027.png" /> generated by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868029.png" /> (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868030.png" /> is a group without centre (an adjoint group), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868031.png" /> is the lattice

<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/l058680/l05868032.png" /></td> </tr></table>

In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868033.png" />. For any additive subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868034.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868035.png" /> there is a unique (up to isomorphism) connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868036.png" /> with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868038.png" />. The centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868039.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868040.png" />, and for the [[Fundamental group|fundamental group]] one has:

<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/l058680/l05868041.png" /></td> </tr></table>

The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868042.png" /> (the centre of the simply-connected Lie group with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868043.png" />) is finite and for the different types of simple Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868044.png" /> it has the following form:''''''<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/l058680/l05868045.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868046.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868047.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868048.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868049.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868050.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868051.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868052.png" /></td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868055.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/l058680/l05868056.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868057.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868058.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868059.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868060.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868061.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868062.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868063.png" /></td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868064.png" /></td> </tr> </tbody> </table>

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

The order of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868065.png" /> is the same as the number of vertices with coefficient 1 in the extended [[Dynkin diagram|Dynkin diagram]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868066.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868067.png" /> of the simply-connected Lie group corresponding to a semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868068.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868069.png" /> can be calculated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868070.png" /> be the [[Cartan decomposition|Cartan decomposition]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868071.png" /> is a maximal compact subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868073.png" /> is its orthogonal complement with respect to the Killing form, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868074.png" /> be the corresponding involutive automorphism, extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868076.png" /> the Cartan subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868077.png" /> containing a Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868079.png" /> an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868080.png" /> that coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868081.png" /> on the roots with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868082.png" /> and extended to the root vectors in an appropriate way, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868083.png" /> the Cartan decomposition of the real form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868084.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868085.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868086.png" /> (see [[#References|[3]]], where this group is calculated for all types of simple algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868087.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868088.png" />).

Every complex semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868089.png" /> has the unique structure of an affine algebraic group compatible with the analytic structure specified on it, and any analytic homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868090.png" /> to an algebraic group is rational. The corresponding algebra of regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868091.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868092.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868093.png" /> is a semi-simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868094.png" />, then in the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868095.png" /> of the simply-connected group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868096.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868097.png" /> there is a smallest subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868098.png" />, called the linearizer, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868099.png" /> is isomorphic to a linear semi-simple Lie group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680100.png" /> is the compact real form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680101.png" />, then

<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/l058680/l058680102.png" /></td> </tr></table>

(see [[#References|[3]]], where this group is calculated for all types of simple Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680103.png" />).

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680104.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680105.png" /> reductions of semi-simple Lie groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680106.png" /> 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>