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The complex Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c0242004.png" /> that is the [[Tensor product|tensor product]] of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c0242005.png" /> with the complex field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c0242006.png" /> over the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c0242007.png" />:
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The ''complexification of a Lie algebra
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$\def\fg{ {\mathfrak g}}\fg$ over $\R$''
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is a complex Lie algebra $\fg_\C$ that is the
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[[Tensor product|tensor product]] of the algebra $\fg$ with the complex field $\C$ over the field of real numbers $\R$:
  
<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/c/c024/c024200/c0242008.png" /></td> </tr></table>
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$$\fg_\C=\fg\otimes_\R \C.$$
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Thus, the complexification of the Lie algebra $\fg$ is obtained from $\fg$ by extending the field of scalars from $\R$ to $\C$. As elements of the algebra $\fg_\C$ one can consider pairs $(u,v)$, $u,v\in\fg$; the operations in $\fg_\C$ are then defined by the formulas:
  
Thus, the complexification of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c0242009.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420010.png" /> by extending the field of scalars from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420012.png" />. As elements of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420013.png" /> one can consider pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420015.png" />; the operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420016.png" /> are then defined by the formulas:
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$$(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2),$$
  
<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/c/c024/c024200/c02420017.png" /></td> </tr></table>
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$$\def\a{\alpha}\a+i\def\b{\beta}\b = (\a u - \b v,\a v + \b u)\ \textrm{ for any } \a,\b\in \R,$$
  
<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/c/c024/c024200/c02420018.png" /></td> </tr></table>
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$$[(u_1,v_1),(u_2,v_2)] = [u_1,u_2] - [v_1,v_2].[v_1,u_2]+[u_1,v_2]).$$
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The algebra $\fg_\C$ is also called the complex hull of the Lie algebra $\fg$.
  
<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/c/c024/c024200/c02420019.png" /></td> </tr></table>
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Certain important properties of an algebra are preserved under complexification: $\fg_\C$ is nilpotent, solvable or semi-simple if and only if $\fg$ has this property. However, simplicity of $\fg$ does not, in general, imply that of $\fg_\C$.
  
The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420020.png" /> is also called the complex hull of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420021.png" />.
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The notion of the complexification of a Lie algebra is closely related to that of a real form of a complex Lie algebra (cf.
 
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[[Form of an (algebraic) structure|Form of an (algebraic) structure]]). A real Lie subalgebra $\def\ff{ {\mathfrak f}}\ff$ of a complex Lie algebra $\def\fh{ {\mathfrak h}}\fh$ is called a real form of $\fh$ if each element $\x\in\fh$ is uniquely representable in the form $x=u+iv$, where $u,v\in\ff$. The complexification of $\ff$ is naturally isomorphic to $\fh$. Not every complex Lie algebra has a real form. On the other hand, a given complex Lie algebra may, in general, have several non-isomorphic real forms. Thus, the Lie algebra of all real matrices of order $n$ and the Lie algebra of all anti-Hermitian matrices of order $n$ are non-isomorphic real forms of the Lie algebra of all complex matrices of order $n$ (which also has other real forms).
Certain important properties of an algebra are preserved under complexification: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420022.png" /> is nilpotent, solvable or semi-simple if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420023.png" /> has this property. However, simplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420024.png" /> does not, in general, imply that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420025.png" />.
 
 
 
The notion of the complexification of a Lie algebra is closely related to that of a real form of a complex Lie algebra (cf. [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). A real Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420026.png" /> of a complex Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420027.png" /> is called a real form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420028.png" /> if each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420029.png" /> is uniquely representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420031.png" />. The complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420032.png" /> is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420033.png" />. Not every complex Lie algebra has a real form. On the other hand, a given complex Lie algebra may, in general, have several non-isomorphic real forms. Thus, the Lie algebra of all real matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420034.png" /> and the Lie algebra of all anti-Hermitian matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420035.png" /> are non-isomorphic real forms of the Lie algebra of all complex matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024200/c02420036.png" /> (which also has other real forms).
 
  
 
====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"> F. Gantmakher,   "On the classification of real simple Lie groups"  ''Mat. Sb.'' , '''5''' :  2  (1939)  pp. 217–250</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ga}}||valign="top"|  F. Gantmakher,  "On the classification of real simple Lie groups"  ''Mat. Sb.'', '''5''' :  2  (1939)  pp. 217–250 
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|-
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|valign="top"|{{Ref|Na}}||valign="top"| M.A. Naimark,  "Theory of
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group representations", Springer  (1982)  (Translated from Russian)
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{{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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Revision as of 22:53, 21 November 2013

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

The complexification of a Lie algebra $\def\fg{ {\mathfrak g}}\fg$ over $\R$ is a complex Lie algebra $\fg_\C$ that is the tensor product of the algebra $\fg$ with the complex field $\C$ over the field of real numbers $\R$:

$$\fg_\C=\fg\otimes_\R \C.$$ Thus, the complexification of the Lie algebra $\fg$ is obtained from $\fg$ by extending the field of scalars from $\R$ to $\C$. As elements of the algebra $\fg_\C$ one can consider pairs $(u,v)$, $u,v\in\fg$; the operations in $\fg_\C$ are then defined by the formulas:

$$(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2),$$

$$\def\a{\alpha}\a+i\def\b{\beta}\b = (\a u - \b v,\a v + \b u)\ \textrm{ for any } \a,\b\in \R,$$

$$[(u_1,v_1),(u_2,v_2)] = [u_1,u_2] - [v_1,v_2].[v_1,u_2]+[u_1,v_2]).$$ The algebra $\fg_\C$ is also called the complex hull of the Lie algebra $\fg$.

Certain important properties of an algebra are preserved under complexification: $\fg_\C$ is nilpotent, solvable or semi-simple if and only if $\fg$ has this property. However, simplicity of $\fg$ does not, in general, imply that of $\fg_\C$.

The notion of the complexification of a Lie algebra is closely related to that of a real form of a complex Lie algebra (cf. Form of an (algebraic) structure). A real Lie subalgebra $\def\ff{ {\mathfrak f}}\ff$ of a complex Lie algebra $\def\fh{ {\mathfrak h}}\fh$ is called a real form of $\fh$ if each element $\x\in\fh$ is uniquely representable in the form $x=u+iv$, where $u,v\in\ff$. The complexification of $\ff$ is naturally isomorphic to $\fh$. Not every complex Lie algebra has a real form. On the other hand, a given complex Lie algebra may, in general, have several non-isomorphic real forms. Thus, the Lie algebra of all real matrices of order $n$ and the Lie algebra of all anti-Hermitian matrices of order $n$ are non-isomorphic real forms of the Lie algebra of all complex matrices of order $n$ (which also has other real forms).

References

[Ga] F. Gantmakher, "On the classification of real simple Lie groups" Mat. Sb., 5 : 2 (1939) pp. 217–250
[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 algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complexification_of_a_Lie_algebra&oldid=30727
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article