U(p,q) - Knowlpedia

For nonnegative integers $p,q$ with $p+q\ge1$ , $\operatorname{U}(p,q)$ is the real Lie group

\operatorname{U}(p,q)=\{A\in\mathrm{GL}(p+q,\mathbb C):A^*I_{p,q}A=I_{p,q}\},\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 $(p+q)^2$ . The definition specifies the full matrix group. It is connected.

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

References

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