SL(3,C) - Knowlpedia

$\operatorname{SL}(3,\mathbb C)$ is the determinant-one special linear group

\operatorname{SL}(3,\mathbb C)=\{A\in M_{3}(\mathbb C):\det A=1\},

with matrix multiplication and the inherited complex Lie group structure.

Dimensions and structure

This group has real dimension $16$ , complex dimension $8$ . Differentiating the determinant at the identity gives the trace-zero tangent algebra. The center consists of scalar matrices $\lambda I$ with $\lambda^{3}=1$ . It is connected and simply connected.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

The inclusion sends a determinant-one matrix to the same invertible matrix; it is injective and smooth and holomorphic. See $\operatorname{GL}(3,\mathbb C)$ .

The scalar-center quotient has kernel $\{\lambda I:\lambda^{3}=1\}$ . See $\operatorname{PSL}(3,\mathbb C)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.