GL(1,R) - Knowlpedia

$\operatorname{GL}(1,\mathbb R)$ is the general linear group

\operatorname{GL}(1,\mathbb R)=\{A\in M_{1}(\mathbb R):\det A\ne0\},

with matrix multiplication and its usual real smooth manifold structure.

Dimensions and structure

This group has real dimension $1$ . It is open in the space of all matrices. Its tangent algebra consists of all matrices with the commutator bracket. The sign of the determinant distinguishes its two connected components. For one-by-one matrices, multiplication identifies this with $\mathbb R^\times$ .

The projectivization map has kernel all nonzero scalar matrices. See $\operatorname{PGL}(1,\mathbb R)$ .

Map a nonzero scalar to its one-by-one matrix; multiplication and inversion coincide. See $\operatorname{GL}(1,\mathbb R)$ .

References

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