Standard Model Lie algebra

Definition

The Standard Model Lie algebra in its compact real form is

\mathfrak g_{\mathrm{SM},\mathbb R} =\mathfrak u(1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(3).

Its complexification is

\mathfrak g_{\mathrm{SM}} =\mathbb C\oplus\mathfrak{sl}_2(\mathbb C)\oplus\mathfrak{sl}_3(\mathbb C).

Authors working with complex representations often denote the second algebra simply by $\mathfrak g_{\mathrm{SM}}$ , so the ground field should be stated.

Relation to the global group

Both

U(1)\times SU(2)\times SU(3) \quad\text{and}\quad \bigl(U(1)\times SU(2)\times SU(3)\bigr)/\mathbb Z_6

have the same real Lie algebra, because a finite central quotient does not change infinitesimal data. Thus the Lie algebra cannot distinguish the two global forms of the Standard Model gauge group.

Complexified matrix realization inside

\mathfrak{sl}_5(\mathbb C)

Differentiating the Georgi–Glashow map and then complexifying gives

(t,X,Y)\longmapsto \begin{pmatrix} 3tI_2+X&0\\ 0&-2tI_3+Y \end{pmatrix},

where $t\in\mathbb C$ , $X\in\mathfrak{sl}_2(\mathbb C)$ , and $Y\in\mathfrak{sl}_3(\mathbb C)$ . Its trace is $6t-6t=0$ , so the image lies in $\mathfrak{sl}_5(\mathbb C)$ .

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, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§1–2.