SL(n,C) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{SL}(n,\mathbb C)$ is the determinant-one special linear group

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

with matrix multiplication and the inherited complex Lie group structure.

Dimensions and structure

This group has real dimension $2(n^2-1)$ , complex dimension $n^2-1$ . Differentiating the determinant at the identity gives the trace-zero tangent algebra. The center consists of scalar matrices $\lambda I$ with $\lambda^{n}=1$ . 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 $\operatorname{GL}(n,\mathbb C)$ .

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

References

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