Knowlpedia

Definition

U(1,1)

U(1,1) as a separately identified real Lie group, with its defining operations and dimension.

U⁡(1,1)\operatorname{U}(1,1) is the real

U⁡(1,1)={A∈GL(2,C):A∗I1,1A=I1,1},I1,1=diag⁡(−I1,I1).\operatorname{U}(1,1)=\{A\in\mathrm{GL}(2,\mathbb C):A^*I_{1,1}A=I_{1,1}\},\qquad I_{1,1}=\operatorname{diag}(-I_{1},I_{1}).

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

Dimensions and structure

This group has real dimension 44. 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.