Knowlpedia

Definition

SU(p,q)

SU(p,q) as a separately identified real Lie group, with its defining operations and dimension.

For nonnegative integers p,qp,q with p+q≥1p+q\ge1, SU⁡(p,q)\operatorname{SU}(p,q) is the real

SU⁡(p,q)={A∈GL(p+q,C):A∗Ip,qA=Ip,q, det⁡A=1},Ip,q=diag⁡(−Ip,Iq).\operatorname{SU}(p,q)=\{A\in\mathrm{GL}(p+q,\mathbb C):A^*I_{p,q}A=I_{p,q},\ \det A=1\},\qquad I_{p,q}=\operatorname{diag}(-I_{p},I_{q}).

The first parameter counts negative directions. The adjoint is conjugate transpose, so the form is Hermitian.

Dimensions and structure

This group has real dimension (p+q)2−1(p+q)^2-1. The definition specifies the full matrix group. It is connected.

The scalar field of its matrix entries does not assign a structure to this real Lie group. In particular, a is not called linear merely because the group has a matrix representation.

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