The isomorphism Spin(6) ≅ SU(4)

Statement

There is an isomorphism of compact Lie groups

\operatorname{Spin}(6)\cong SU(4).

Infinitesimally it gives an isomorphism of real Lie algebras

\mathfrak{spin}(6)\cong\mathfrak{so}(6) \cong\mathfrak{su}(4),

and after complexification it is the exceptional root-system identification $D_3=A_3$ :

\mathfrak{so}_6(\mathbb C)\cong\mathfrak{sl}_4(\mathbb C).

Corresponding representations

Under a choice of this isomorphism, the two complex half-spin representations of $\operatorname{Spin}(6)$ correspond to the defining representation of $SU(4)$ on $\mathbb C^4$ and its dual:

\Delta^+\leftrightarrow\mathbb C^4, \qquad \Delta^-\leftrightarrow(\mathbb C^4)^*.

Interchanging chirality interchanges the two right-hand representations.

The complexification of the six-dimensional vector representation corresponds to the second exterior power:

\mathbb C^6\cong\Lambda^2\mathbb C^4.

Because $SU(4)$ preserves a Hermitian form and a complex volume form on $\mathbb C^4$ , $\Lambda^2\mathbb C^4$ carries an invariant real structure. Its fixed real subspace has dimension six and gives the homomorphism

SU(4)\longrightarrow SO(6).

Its kernel is $\{\pm I\}$ , so it is the spin double covering.

Why the group isomorphism follows

The compact groups $\operatorname{Spin}(6)$ and $SU(4)$ are both connected and simply connected, and their Lie algebras are isomorphic. A Lie-algebra isomorphism between connected simply connected Lie groups integrates uniquely to a Lie-group isomorphism.

This is stronger than $SO(6)\cong SU(4)$ : that latter statement is false. Instead,

SO(6)\cong SU(4)/\{\pm I\}.

The remaining central elements $\pm iI$ act nontrivially in the vector-covering construction before quotienting by the appropriate kernel.

Convention warning

No preferred isomorphism singles out which of $\Delta^+$ or $\Delta^-$ is $\mathbb C^4$ . Orientation reversal on the orthogonal side and complex conjugation on the unitary side exchange the two choices.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter I, §5. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§19–20. Publisher record.