Complex Lie algebra so10(C)

The complex Lie algebra $\mathfrak{so}_{10}(\mathbb C)$ consists of endomorphisms preserving a nondegenerate symmetric bilinear form infinitesimally:

\mathfrak{so}_{10}(\mathbb C)=\{X\in M_{10}(\mathbb C):X^{\mathsf T}+X=0\}

after choosing the standard form. It is simple, of complex dimension $45$ , rank $5$ , and Dynkin type $D_5$ .

Its distinguished irreducible modules include the vector module $\mathbf{10}$ , the two inequivalent half-spin modules $\mathbf{16}_+$ and $\mathbf{16}_-$ , and the adjoint module $\mathbf{45}\cong\Lambda^2\mathbf{10}$ .

Spin representations and global groups

The half-spin modules are representations of the spin group $\operatorname{Spin}(10,\mathbb C)$ , the simply connected complex group with Lie algebra $\mathfrak{so}_{10}(\mathbb C)$ . They do not descend to the ordinary matrix group $SO(10,\mathbb C)$ . Thus the notation “ $SO(10)$ grand unified theory” commonly refers globally to $\operatorname{Spin}(10)$ when a $\mathbf{16}$ of fermions is present.

The compact real form integrates to compact $\operatorname{Spin}(10)$ ; the split real form is $\mathfrak{so}(5,5)$ . Other real forms, such as $\mathfrak{so}(p,10-p)$ , have the same complexification but different real-group representation theory.

Role in the E-series chain

The regular inclusion

\mathfrak{sl}_5\subset\mathfrak{so}_{10}

is followed by the inclusion of $\mathfrak{so}_{10}$ in the exceptional Lie algebra $\mathfrak e_6$ . Under $\mathfrak{so}_{10}\oplus\mathbb C$ , the adjoint representation of $\mathfrak e_6$ branches, up to a choice of charge sign, as

\mathbf{78}\cong\mathbf{45}_0\oplus\mathbf1_0 \oplus\mathbf{16}_{3}\oplus\mathbf{16}^{*}_{-3}.

This places $\mathfrak{so}_{10}$ at the $E_5$ stage of the $E_n$ series used in the three-generation construction.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plate IV. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Sections 19--20. Publisher record.
John C. Baez and John Huerta, The Algebra of Grand Unified Theories, Bulletin of the American Mathematical Society 47 (2010), 483--552. DOI.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.