Difference between revisions of "Adjoint representation of a Lie group"
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''or algebraic group $G$'' | ''or algebraic group $G$'' | ||
| − | The linear representation $\def\Ad{\textrm{Ad}\;} \Ad | + | The ''adjoint representation'' of a Lie group $G$ is |
| + | the linear representation $\def\Ad{\textrm{Ad}\;} \Ad $ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}\g$ of $G$), given by the mapping which sends 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 of isomorphisms of a vector space $V$, then | ||
$$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$ | $$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$ | ||
| − | 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 kernel $\ker \Ad$ contains the centre of $G$, and if $G$ is connected and if the ground field has characteristic zero, it 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 $\g$, defined as follows: | ||
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 | 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 | ||
$$(\ad x)y = [x,y],\quad x,y\in \g,$$ | $$(\ad x)y = [x,y],\quad x,y\in \g,$$ | ||
| − | 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. | + | 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==== | ||
Latest revision as of 16:57, 27 March 2012
2020 Mathematics Subject Classification: Primary: 17-XX Secondary: 22-XX [MSN][ZBL]
or algebraic group $G$
The adjoint representation of a Lie group $G$ is the linear representation $\def\Ad{\textrm{Ad}\;} \Ad $ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}\g$ of $G$), given by the mapping which sends 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 of isomorphisms of a vector 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, it 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 $\g$, defined as follows:
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 |
Adjoint representation of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_representation_of_a_Lie_group&oldid=23563