Special block unitary group

Definition

For positive integers $p,q$ , the special block unitary group is

S(U(p)\times U(q)) =\{(A,B)\in U(p)\times U(q):\det(A)\det(B)=1\}.

The map $(A,B)\mapsto\operatorname{diag}(A,B)$ identifies it with the subgroup of $SU(p+q)$ preserving $\mathbb C^{p+q}=\mathbb C^p\oplus\mathbb C^q$ .

Lie algebra

Its Lie algebra is

\mathfrak s\bigl(\mathfrak u(p)\oplus\mathfrak u(q)\bigr) =\{(X,Y)\in\mathfrak u(p)\oplus\mathfrak u(q): \operatorname{tr}X+\operatorname{tr}Y=0\}.

It is isomorphic as a real Lie algebra to $\mathfrak u(1)\oplus\mathfrak{su}(p)\oplus\mathfrak{su}(q)$ , although the corresponding connected Lie group need not be the direct product.

Central-quotient presentation

The surjective homomorphism

U(1)\times SU(p)\times SU(q)\longrightarrow S(U(p)\times U(q)), \qquad (z,A,B)\longmapsto (z^qA,z^{-p}B)

has kernel

\{(z,z^{-q}I_p,z^pI_q):z^{pq}=1\}\cong\mathbb Z_{pq}.

Hence

S(U(p)\times U(q))\cong \bigl(U(1)\times SU(p)\times SU(q)\bigr)/\mathbb Z_{pq}.

For $p=2,q=3$ , the map is

(z,A,B)\longmapsto(z^3A,z^{-2}B),

with kernel $\{(z,z^{-3}I_2,z^2I_3):z^6=1\}$ . Thus

S(U(2)\times U(3))\cong \bigl(U(1)\times SU(2)\times SU(3)\bigr)/\mathbb Z_6.

Standard Model role

The group $S(U(2)\times U(3))$ is the image of the Georgi–Glashow homomorphism inside $SU(5)$ . It is one common convention for the effective Standard Model gauge group.

References

John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv record. Relevant: §3.1.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv record. Relevant: §§1 and 3.