paper
The Algebra of Grand Unified Theories
John C. Baez · John Huerta
Use in this reference: Linked primary source supporting original derivations and calculations. No upstream text, software, or representation dataset is redistributed.
Relevant material: Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| su(3) ⊕ su(2) ⊕ u(1) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Real dimension | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Rank | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Lie-algebra center | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Simple | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Global groups and charge normalization | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Color algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Weak algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Abelian algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 1 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 2 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 3 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 6 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Real dimension | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Rank | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Compact | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Connected | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Fundamental group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Specified quotient order | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Abstract center | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Topology and the universal cover | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Hypercharge and SU(5) | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | The common gauge algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 2 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 3 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 6 | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Real dimension | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Rank | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Compact | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Connected | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Fundamental group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Specified quotient order | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Abstract center | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Topology and the universal cover | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Hypercharge and SU(5) | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | The common gauge algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(2) | Specified compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Real dimension | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Rank | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Compact | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Connected | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Fundamental group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Specified quotient order | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Abstract center | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Topology and the universal cover | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Hypercharge and SU(5) | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | The common gauge algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(3) | Specified compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Real dimension | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Rank | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Compact | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Connected | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Fundamental group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Specified quotient order | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Abstract center | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Matter and the embedded kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Topology and the universal cover | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Hypercharge and SU(5) | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | The common gauge algebra | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Specified compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) | Block subgroup of SU(5) with the specified map | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Injective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Surjective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Matter descent | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(2) | Chosen global form | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Injective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Surjective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Matter descent | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(3) | Chosen global form | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Injective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Surjective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Matter descent | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Compact covering product | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(1) → G_SM^(6) | Chosen global form | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Image | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Injective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Surjective onto SU(5) | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Physical hypercharge | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Keep the charge | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | From homomorphism to embedding | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Underlying compact product, with the X coordinate stated here | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Parent group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| SU(3) × SU(2) × U(1)_X → SU(5) | Image after converting X to Q=−X | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Definition | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Kernel | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Injective | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Image | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | What the embedding establishes | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | The specified diagonal quotient | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Parent compact group | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |
| G_SM^(6) ↪ SU(5) | Computable block map in the X convention | Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly. |