Knowlpedia

Definition

Projective general linear Lie group

The real or complex general linear Lie group modulo its closed central subgroup of scalar matrices.

Definition

Let F=R\mathbb F=\mathbb R or C\mathbb C, and let n≥2n\ge2. The projective general linear Lie group is the

PGL⁡n(F)=GL⁡n(F)/(F×In).\operatorname{PGL}_n(\mathbb F) =\operatorname{GL}_n(\mathbb F)/(\mathbb F^\times I_n).

The scalar subgroup is closed and central. Therefore the abstract carries the unique quotient manifold structure for which the projection is a .

Dimension and Lie algebra

The scalar subgroup has dimension 11 over F\mathbb F, so

dim⁡RPGL⁡n(R)=n2−1,dim⁡CPGL⁡n(C)=n2−1.\dim_{\mathbb R}\operatorname{PGL}_n(\mathbb R)=n^2-1, \qquad \dim_{\mathbb C}\operatorname{PGL}_n(\mathbb C)=n^2-1.

The complex group consequently has underlying real dimension 2(n2−1)2(n^2-1). Its is

pgln(F)=gln(F)/(FIn).\mathfrak{pgl}_n(\mathbb F) =\mathfrak{gl}_n(\mathbb F)/(\mathbb F I_n).

Because F\mathbb F has characteristic zero, X↦X−tr⁡XnInX\mapsto X-\frac{\operatorname{tr}X}{n}I_n identifies this quotient with sln(F)\mathfrak{sl}_n(\mathbb F).

Connected components over R\mathbb R and C\mathbb C

The PGL⁡n(C)\operatorname{PGL}_n(\mathbb C) is connected. For the real group, scalar multiplication changes determinant by λn\lambda^n. It follows that

π0(PGL⁡n(R))≅{1,n odd,Z/2,n even.\pi_0(\operatorname{PGL}_n(\mathbb R))\cong \begin{cases} 1,&n\ \text{odd},\\ \mathbb Z/2,&n\ \text{even}. \end{cases}

Indeed, for odd nn, a negative scalar identifies the two determinant-sign components of GL⁡n(R)\operatorname{GL}_n(\mathbb R); for even nn, scalar matrices always have positive determinant and the sign survives the quotient.

The identity component is the image of SL⁡n(R)\operatorname{SL}_n(\mathbb R), namely PSL⁡n(R)\operatorname{PSL}_n(\mathbb R). Thus it is the whole group for odd nn and has index 22 for even nn.

References
  1. Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015. Publisher record. Relevant: matrix Lie groups, quotient Lie groups, and classical Lie algebras.
  2. James S. Milne, Lie Algebras, Algebraic Groups, and Lie Groups, 2013. Author-maintained course notes. Relevant: classical groups and their Lie algebras.