Adjoint representation of a Lie group

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