SL(n,R) - Knowlpedia

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

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

with matrix multiplication and the inherited real Lie group structure.

Dimensions and structure

This group has real 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 for every $n$ . For $n\ge2$ , it is noncompact and is not simply connected, distinguishing it from the compact special unitary group. For $n=1$ , it is the trivial group and is therefore compact and simply connected.

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

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

References

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