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| − | ''or algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108801.png" />''
| + | {{MSC|17|22}} |
| | + | {{TEX|done}} |
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| − | The linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108802.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108803.png" /> in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108804.png" /> (or in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108805.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108806.png" />) mapping each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108807.png" /> to the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108808.png" /> of the inner automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108809.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088010.png" /> is a linear group in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088011.png" />, then
| + | ''or algebraic group $G$'' |
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| − | <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/a/a010/a010880/a01088012.png" /></td> </tr></table>
| + | The linear representation $\def\Ad{\textrm{Ad}\;} \Ad G$ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}$ of $G$) mapping each $a\in G$ to the differential $\def\Int{\textrm{Int}\;}\Ad a = d(\Int a)_e$ of the inner automorphism $\Int a: x\mapsto axa^{-1}$. If $G\subseteq \def\GL{\textrm{GL}} \GL(V)$ is a linear group in a space $V$, then |
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| − | The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088013.png" /> contains the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088014.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088015.png" /> is connected and if the ground field has characteristic zero, coincides with this centre. The differential of the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088016.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088017.png" /> coincides with the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088019.png" />.
| + | $$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$ |
| | + | The kernel $\ker \Ad$ contains the centre of $G$, and if $G$ is connected and if the ground field has characteristic zero, coincides with this centre. The differential of the adjoint representation of $\def\ad{\textrm{ad}\;} G$ at $e$ coincides with the adjoint representation $\ad$ of $\g$. |
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| − | The adjoint representation of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088020.png" /> is the linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088021.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088022.png" /> into the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088023.png" /> acting by the formula | + | The adjoint representation of a Lie algebra $\g$ is the linear representation $\ad$ of the algebra $\g$ into the module $\g$ acting by the formula |
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| − | <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/a/a010/a010880/a01088024.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088025.png" /> is the bracket operation in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088026.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088027.png" /> is the centre of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088028.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088029.png" /> are derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088030.png" /> and are called inner derivations. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088031.png" /> is called the adjoint linear Lie algebra and is an ideal in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088032.png" /> of all derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088033.png" />, moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088034.png" /> is the one-dimensional cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088036.png" />, defined by the adjoint representation. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088038.png" /> is a semi-simple Lie algebra over a field of characteristic zero.
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
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| − | ====Comments====
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| | + | $$(\ad x)y = [x,y],\quad x,y\in \g,$$ |
| | + | where $[\;,\;]$ is the bracket operation in the algebra $\g$. The kernel $\ker \ad$ is the centre of the Lie algebra $\g$. The operators $\ad x$ are derivations of $\g$ and are called inner derivations. The image $\ad \g$ is called the adjoint linear Lie algebra and is an ideal in the Lie algebra $\def\Der{\textrm{Der}}\Der \g$ of all derivations of $\g$, moreover $\Der \g/\ad\g$ is the one-dimensional cohomology space $H^1(\g,\g)$ of $\g$, defined by the adjoint representation. In particular, $\ad \g = \Der\g$ if $\g$ is a semi-simple Lie algebra over a field of characteristic zero. |
| | | | |
| | ====References==== | | ====References==== |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR></table>
| + | {| |
| | + | |- |
| | + | |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} |
| | + | |- |
| | + | |valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} |
| | + | |- |
| | + | |valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0143793}} {{ZBL|0121.27504}} |
| | + | |- |
| | + | |valign="top"|{{Ref|Po}}||valign="top"| L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} |
| | + | |- |
| | + | |valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} |
| | + | |- |
| | + | |} |
2020 Mathematics Subject Classification: Primary: 17-XX Secondary: 22-XX [MSN][ZBL]
or algebraic group $G$
The linear representation $\def\Ad{\textrm{Ad}\;} \Ad G$ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}$ of $G$) mapping each $a\in G$ to the differential $\def\Int{\textrm{Int}\;}\Ad a = d(\Int a)_e$ of the inner automorphism $\Int a: x\mapsto axa^{-1}$. If $G\subseteq \def\GL{\textrm{GL}} \GL(V)$ is a linear group in a space $V$, then
$$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$
The kernel $\ker \Ad$ contains the centre of $G$, and if $G$ is connected and if the ground field has characteristic zero, coincides with this centre. The differential of the adjoint representation of $\def\ad{\textrm{ad}\;} G$ at $e$ coincides with the adjoint representation $\ad$ of $\g$.
The adjoint representation of a Lie algebra $\g$ is the linear representation $\ad$ of the algebra $\g$ into the module $\g$ acting by the formula
$$(\ad x)y = [x,y],\quad x,y\in \g,$$
where $[\;,\;]$ is the bracket operation in the algebra $\g$. The kernel $\ker \ad$ is the centre of the Lie algebra $\g$. The operators $\ad x$ are derivations of $\g$ and are called inner derivations. The image $\ad \g$ is called the adjoint linear Lie algebra and is an ideal in the Lie algebra $\def\Der{\textrm{Der}}\Der \g$ of all derivations of $\g$, moreover $\Der \g/\ad\g$ is the one-dimensional cohomology space $H^1(\g,\g)$ of $\g$, defined by the adjoint representation. In particular, $\ad \g = \Der\g$ if $\g$ is a semi-simple Lie algebra over a field of characteristic zero.
References
| [Bo] |
N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) MR0682756 Zbl 0319.17002
|
| [Hu] |
J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
|
| [Ja] |
N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979)) MR0143793 Zbl 0121.27504
|
| [Po] |
L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
|
| [Se] |
J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
|