nLab U(2)

Idea

$U(2)$ is the unitary group in dimension 2. It is used to describe almost complex structures on 4-manifolds.

Properties

Proposition

One has exceptional isomorphisms between $U(2)$ and the spinᶜ group in dimension 3 as well as the spinʰ group in dimension 2:

$$U(2) \cong Spin^\mathrm{c}(3) \cong Spin^\mathrm{h}(2).$$

(Gompf & Stipsicz 99, Ex. 2.4.14, Nicolaescu 00, Ex. 1.3.9)

Proof

Using the exceptional isomorphism $Spin(3)\cong SU(2)$ yields:

$$Spin^\mathrm{c}(3) \cong(Spin(3)\times U(1))/\mathbb{Z}_2 =(SU(2)\times U(1))/\mathbb{Z}_2 \cong U(2)$$

Using the exceptional isomorphism $Spin(2)\cong U(1)$ as well as $Sp(1)\cong SU(2)$ yields:

$$Spin^\mathrm{h}(2) \cong(Spin(2)\times Sp(1))/\mathbb{Z}_2 =(SU(2)\times U(1))/\mathbb{Z}_2 \cong U(2)$$

(In general, one has $(SU(n)\times U(1))/\mathbb{Z}_n\cong U(n)$ using the homomorphism theorem? on the group homomorphism $SU(n)\times U(1)\rightarrow U(n),(U,z)\mapsto U \cdot z$, which is surjective and has $\{(\zeta_n^k\mathbf{1}_n,\zeta_n^{-k})|[k]\in\mathbb{Z}_n \}\cong\mathbb{Z}_n$ as kernel.)

Related concepts

• Spinᶜ(4)

• Spinᶜ(6)

References

• Robert Gompf and András Stipsicz, 4-Manifolds and Kirby Calculus (1999), Graduate Studies

  in Mathematics, Volume 20 [ISBN: 978-0-8218-0994-5, doi:10.1090/gsm/020]

• Liviu Nicolaescu, Notes on Seiberg-Witten theory, American Mathematical Society (2000) [ISBN:978-0-8218-2145-9, pdf]