SL(2,R) - Knowlpedia

$\operatorname{SL}(2,\mathbb R)$ is the determinant-one special linear group

\operatorname{SL}(2,\mathbb R)=\{A\in M_{2}(\mathbb R):\det A=1\},

with matrix multiplication and the inherited real Lie group structure.

Dimensions and structure

This group has real dimension $3$ . Differentiating the determinant at the identity gives the trace-zero tangent algebra. The center consists of scalar matrices $\lambda I$ with $\lambda^{2}=1$ . It is connected but is not simply connected for matrix size at least two; its noncompactness distinguishes it from the compact special unitary group.

The inclusion sends a determinant-one matrix to the same invertible matrix; it is injective and smooth. See $\operatorname{GL}(2,\mathbb R)$ .

The scalar-center quotient has kernel $\{\lambda I:\lambda^{2}=1\}$ . See $\operatorname{PSL}(2,\mathbb R)$ .

References

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