Low-dimensional Lie algebra isomorphisms

Core idea

Low-dimensional Lie algebra isomorphisms identify several different matrix constructions after the scalar field is fixed. The identities below are Lie-algebra statements; a corresponding group map can have a nontrivial discrete kernel.

Explicit identifications

The following catalogue arrows retain the defining field:

• $\mathfrak{so}(4)$ and su(2) direct sum with itself: Identify so(4) with bivectors in oriented Euclidean four-space. The self-dual and anti-self-dual bivectors are commuting three-dimensional ideals, each isomorphic to so(3), hence to su(2).
• $\mathfrak{so}(4,\mathbb C)$ and sl(2,C) direct sum with itself: Complexify the compact so(4) decomposition and the real form su(2).
• $\mathfrak{so}(2,2)$ and sl(2,R) direct sum with itself: The action (X,Y)·A=XA−AY on real 2-by-2 matrices preserves determinant, a form of signature (2,2). Its infinitesimal kernel is zero: XA=AY for every A forces X=Y scalar, and trace zero then forces both to vanish. Both algebras have dimension six.
• $\mathfrak{su}(2)$ and sp(1) — compact symplectic Lie algebra: Identify imaginary quaternions with trace-zero skew-Hermitian 2-by-2 matrices; quaternion multiplication is represented by matrix multiplication.
• $\mathfrak{su}(2)$ and so(3,R) — real orthogonal Lie algebra: The adjoint action on the three-dimensional imaginary-quaternion space preserves its Euclidean norm, is faithful infinitesimally, and both algebras have dimension three.
• $\mathfrak{sp}(2)$ and so(5,R) — real orthogonal Lie algebra: Act by X·A=XA−AX on traceless Hermitian quaternionic 2-by-2 matrices. This five-dimensional real space has positive form tr(A²); an element in the kernel commutes with every Hermitian matrix. Commuting with diagonal projections makes it diagonal; commuting with all quaternionic off-diagonal entries makes it a real scalar. Skew-adjointness then forces it to vanish. Source and target both have dimension ten.
• $\mathfrak{su}(4)$ and so(6,R) — real orthogonal Lie algebra: The exterior-square representation preserves the symmetric wedge pairing on Λ²C⁴. Its compatible antilinear Hodge involution has a six-dimensional real fixed space; the differentiated action is faithful and both algebras have dimension fifteen.
• $\mathfrak{sl}(2,\mathbb R)$ and sp(2,R) — real symplectic Lie algebra: For a 2-by-2 matrix X, XᵀJ+JX=(tr X)J.
• $\mathfrak{sl}(2,\mathbb C)$ and sp(2,C) — complex symplectic Lie algebra: The same matrix identity holds over C.
• $\mathfrak{sl}(2,\mathbb C)$ and so(3,C) — complex orthogonal Lie algebra: The adjoint action preserves the nondegenerate Killing form; its kernel is the zero center, and both algebras have dimension three.
• $\mathfrak{sl}(2,\mathbb R)$ and so(2,1) — indefinite orthogonal real Lie algebra: The form tr(X²) on trace-zero real 2-by-2 matrices has signature (2,1). Its adjoint action is injective and dimension three.
• $\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$ and so(3,1) — indefinite orthogonal real Lie algebra: The action X·H=XH+HX* on Hermitian 2-by-2 complex matrices infinitesimally preserves determinant, a quadratic form of signature (1,3). If this action vanishes then X is skew-Hermitian and commutes with every Hermitian matrix, hence is scalar; trace zero forces X=0. Both algebras have real dimension six, and changing the sign of the form does not change its orthogonal Lie algebra.
• $\mathfrak{sp}(4,\mathbb C)$ and so(5,C) — complex orthogonal Lie algebra: Complexify the compact sp(2)–so(5) isomorphism.
• $\mathfrak{sl}(4,\mathbb C)$ and so(6,C) — complex orthogonal Lie algebra: Complexify the compact su(4)–so(6) isomorphism.
• $\mathfrak{so}(1,1)$ and Abelian real Lie algebra of dimension 1: Both algebras have one basis vector and zero bracket.
• $\mathfrak{so}(2)$ and Abelian real Lie algebra of dimension 1: Both algebras have one basis vector and zero bracket, despite different global groups.
• $\mathfrak u(1)$ and Abelian real Lie algebra of dimension 1: The real-linear map it↦t identifies the zero brackets.
• $\mathfrak{sl}(1,\mathbb H)$ and sp(1) — compact symplectic Lie algebra: A quaternion has zero real part exactly when its conjugate is its negative.

Distinguishing scalar restriction

$\mathfrak{sl}(2,\mathbb C)$ has complex dimension three, while its underlying real Lie algebra has real dimension six. It cannot be real-linearly isomorphic to the three-dimensional $\mathfrak{sl}(2,\mathbb R)$. Triality in $\mathfrak{so}(8)$ concerns automorphisms permuting representations; it does not identify the vector and spinor modules without twisting.