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The complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242104.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242105.png" /> as a real Lie subgroup such that the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242106.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242107.png" /> is a real form of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c0242109.png" /> (see [[Complexification of a Lie algebra|Complexification of a Lie algebra]]). One then says that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421010.png" /> is a real form of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421011.png" />. For example, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421012.png" /> of all unitary matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421013.png" /> is a real form of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421014.png" /> of all non-singular matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421015.png" /> with complex entries.
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The ''complexification of a Lie group $G$ over $\R$'' is
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a complex Lie group $G_\C$ containing $G$ as a real Lie subgroup such that the Lie algebra $\def\fg{ {\mathfrak g}}\fg$ of $G$ is a real form of the Lie algebra $\fg_\C$ of $G_\C$ (see
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[[Complexification of a Lie algebra|Complexification of a Lie algebra]]). One then says that the group $G$ is a real form of the Lie group $G_\C$. For example, the group $\def\U{ {\rm U}}\U(n)$ of all unitary matrices of order $n$ is a real form of the group $\def\GL{ {\rm GL}}\GL(n,\C)$ of all non-singular matrices of order $n$ with complex entries.
  
There is a one-to-one correspondence between the complex-analytic linear representations of a connected simply-connected complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421016.png" /> and the real-analytic representations of its connected real form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421017.png" />, under which irreducible representations correspond to each other. This correspondence is set up in the following way: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421018.png" /> is an (irreducible) finite-dimensional complex-analytic representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421019.png" />, then the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421021.png" /> is an (irreducible) real-analytic representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421022.png" />.
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There is a one-to-one correspondence between the complex-analytic linear representations of a connected simply-connected complex Lie group $G_\C$ and the real-analytic representations of its connected real form $G$, under which irreducible representations correspond to each other. This correspondence is set up in the following way: If $\rho$ is an (irreducible) finite-dimensional complex-analytic representation of $G_\C$, then the restriction of $\rho$ to $G$ is an (irreducible) real-analytic representation of $G$.
  
Not every real Lie group has a complexification. In particular, a connected semi-simple Lie group has a complexification if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421023.png" /> is linear, that is, is isomorphic to a subgroup of some group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421024.png" />. For example, the universal covering of the group of real second-order matrices with determinant 1 does not have a complexification. On the other hand, every compact Lie group has a complexification.
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Not every real Lie group has a complexification. In particular, a connected semi-simple Lie group has a complexification if and only if $G$ is linear, that is, is isomorphic to a subgroup of some group $\GL(n,\C)$. For example, the universal covering of the group of real second-order matrices with determinant 1 does not have a complexification. On the other hand, every compact Lie group has a complexification.
  
The non-existence of complexifications for certain real Lie groups inspired the introduction of the more general notion of a universal complexification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421025.png" /> of a real Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421026.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421027.png" /> is a complex Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421028.png" /> is a real-analytic homomorphism such that for every complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421029.png" /> and every real-analytic homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421030.png" /> there exists a unique complex-analytic homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421032.png" />. The universal complexification of a Lie group always exists and is uniquely defined [[#References|[3]]]. Uniqueness means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421033.png" /> is another universal complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421034.png" />, then there is a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421036.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421037.png" />, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421038.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421039.png" /> and the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024210/c02421040.png" /> is discrete.
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The non-existence of complexifications for certain real Lie groups inspired the introduction of the more general notion of a universal complexification $(\tilde G,\tau)$ of a real Lie group $G$. Here $\tilde G$ is a complex Lie group and $\tau : G\to \tilde G$ is a real-analytic homomorphism such that for every complex Lie group $H$ and every real-analytic homomorphism $\alpha : G\to H$ there exists a unique complex-analytic homomorphism $\beta : \tilde G\to H$ such that $\alpha=\beta\circ \tau$. The universal complexification of a Lie group always exists and is uniquely defined {{Cite|Bo}}. Uniqueness means that if $(\tilde G',\tau')$ is another universal complexification of $G$, then there is a natural isomorphism $\lambda : \tilde G\to \tilde G'$ such that $\lambda\circ\tau = \tau'$. In general, $\dim_\C\;\tilde G \le \dim_\R G$, but if $G$ is simply connected, then $\dim_\C \tilde G = \dim_\R G$ and the kernel of $\tau$ is discrete.
  
See also [[Form of an algebraic group|Form of an algebraic group]].
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See also
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[[Form of an algebraic group|Form of an algebraic group]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Naimark,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>
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|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley  (1975)  (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}}
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|-
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|valign="top"|{{Ref|Na}}||valign="top"| M.A. Naimark,  "Theory of group representations", Springer  (1982)  (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}}
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|-
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|valign="top"|{{Ref|Zh}}||valign="top"| D.P. Zhelobenko,  "Compact Lie groups and their representations", Amer. Math. Soc.  (1973)  (Translated from Russian)  {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}}
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Latest revision as of 15:05, 13 April 2024

2020 Mathematics Subject Classification: Primary: 22E [MSN][ZBL]

The complexification of a Lie group $G$ over $\R$ is a complex Lie group $G_\C$ containing $G$ as a real Lie subgroup such that the Lie algebra $\def\fg{ {\mathfrak g}}\fg$ of $G$ is a real form of the Lie algebra $\fg_\C$ of $G_\C$ (see Complexification of a Lie algebra). One then says that the group $G$ is a real form of the Lie group $G_\C$. For example, the group $\def\U{ {\rm U}}\U(n)$ of all unitary matrices of order $n$ is a real form of the group $\def\GL{ {\rm GL}}\GL(n,\C)$ of all non-singular matrices of order $n$ with complex entries.

There is a one-to-one correspondence between the complex-analytic linear representations of a connected simply-connected complex Lie group $G_\C$ and the real-analytic representations of its connected real form $G$, under which irreducible representations correspond to each other. This correspondence is set up in the following way: If $\rho$ is an (irreducible) finite-dimensional complex-analytic representation of $G_\C$, then the restriction of $\rho$ to $G$ is an (irreducible) real-analytic representation of $G$.

Not every real Lie group has a complexification. In particular, a connected semi-simple Lie group has a complexification if and only if $G$ is linear, that is, is isomorphic to a subgroup of some group $\GL(n,\C)$. For example, the universal covering of the group of real second-order matrices with determinant 1 does not have a complexification. On the other hand, every compact Lie group has a complexification.

The non-existence of complexifications for certain real Lie groups inspired the introduction of the more general notion of a universal complexification $(\tilde G,\tau)$ of a real Lie group $G$. Here $\tilde G$ is a complex Lie group and $\tau : G\to \tilde G$ is a real-analytic homomorphism such that for every complex Lie group $H$ and every real-analytic homomorphism $\alpha : G\to H$ there exists a unique complex-analytic homomorphism $\beta : \tilde G\to H$ such that $\alpha=\beta\circ \tau$. The universal complexification of a Lie group always exists and is uniquely defined [Bo]. Uniqueness means that if $(\tilde G',\tau')$ is another universal complexification of $G$, then there is a natural isomorphism $\lambda : \tilde G\to \tilde G'$ such that $\lambda\circ\tau = \tau'$. In general, $\dim_\C\;\tilde G \le \dim_\R G$, but if $G$ is simply connected, then $\dim_\C \tilde G = \dim_\R G$ and the kernel of $\tau$ is discrete.

See also Form of an algebraic group.

References

[Bo] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[Na] M.A. Naimark, "Theory of group representations", Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[Zh] D.P. Zhelobenko, "Compact Lie groups and their representations", Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
How to Cite This Entry:
Complexification of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complexification_of_a_Lie_group&oldid=14697
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article