Conjugation on SL(2,C) distinguishes real and complex group maps

For $G=SL(2,\mathbb C)$ , entrywise complex conjugation

c:G\longrightarrow G,\qquad c(A)=\overline A,

is an automorphism in the real Lie-group category but is not a morphism in the complex Lie-group category. It is smooth, multiplicative, and involutive, but antiholomorphic rather than holomorphic.

Direct verification

Conjugation respects determinant and matrix multiplication, so it maps $G$ to itself and obeys $c(AB)=c(A)c(B)$ . It is its own inverse. Its differential at the identity is $X\mapsto\overline X$ on $\mathfrak{sl}_2(\mathbb C)$ , which is real-linear but sends $iX$ to $-i\overline X$ . For nonzero $X$ , this fails complex linearity, so the original map is not holomorphic.

Other recorded maps

For each $B\in GL_2(\mathbb C)$ , the map $A\mapsto BAB^{-1}$ is a holomorphic automorphism of $G$ . Composing it with $c$ supplies antiholomorphic real automorphisms. The constant map $A\mapsto I_2$ is a group endomorphism in both categories and is not an automorphism.

These formulas provide families of maps. This knowl does not claim a classification of every abstract group endomorphism or every real or holomorphic group endomorphism; the associated catalogue records are marked partial.

Where linear endomorphisms live

$SL(2,\mathbb C)$ is not closed under addition or scalar multiplication. Real-linear versus complex-linear endomorphisms therefore require a specified vector space, such as its Lie algebra $\mathfrak{sl}_2(\mathbb C)$ or the ambient matrix algebra. A linear map of the ambient space need not restrict to a group homomorphism.

See the explicit linear-endomorphism comparison for its Lie algebra.