GL(1,C) - Knowlpedia

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

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

with matrix multiplication and its usual complex analytic manifold structure.

Dimensions and structure

This group has real dimension $2$ , complex dimension $1$ . It is open in the space of all matrices. Its tangent algebra consists of all matrices with the commutator bracket. It is connected; its maximal compact subgroup is the unitary group of the same matrix size. For one-by-one matrices, multiplication identifies this with $\mathbb C^\times$ .

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

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

References

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