Math 2.0

Topological groups, Lie groups Classical Lie groups Example

Exceptional Lie group G2

Builds on OctonionsSpecial orthogonal groupSpecial unitary group SU(n)

Example

Let 𝕆\mathbb{O} be the octonions, the 88-dimensional real normed division algebra. The compact Lie group G2G_2 is the automorphism group of 𝕆\mathbb{O}: G2={g∈GL(𝕆):g(xy)=g(x) g(y) for all x,y∈𝕆}.G_2 = \set{g \in GL(\mathbb{O}) : g(xy) = g(x)\, g(y) \text{ for all } x, y \in \mathbb{O}}. It is a closed subgroup of GL8(ℝ)GL_8(\R), since it is cut out by polynomial equations, so it is a Lie group. Each g∈G2g \in G_2 fixes 11 and preserves the octonion norm, so it maps the 77-dimensional space Im⁡𝕆\operatorname{Im} \mathbb{O} of imaginary octonions to itself by a rotation. This makes G2G_2 a closed subgroup of SO(7), hence compact.

G2G_2 is a connected and simply connected simple Lie group of dimension 1414 and rank 22, with trivial centre.

Remarks

G2G_2 is the smallest of the five exceptional compact simple Lie groups G2,F4,E6,E7,E8G_2, F_4, E_6, E_7, E_8. Its Dynkin diagram has two nodes joined by a triple edge, and its root system has 1212 roots, of two lengths whose squares have ratio 33. It acts transitively on the unit sphere S6S^6 of Im⁡𝕆\operatorname{Im} \mathbb{O}, with stabilizer isomorphic to SU(3), so G2/SU(3)≅S6G_2 / SU(3) \cong S^6. The name G2G_2 also denotes the complex simple Lie group of the same type and its noncompact split real form. Riemannian 77-manifolds whose holonomy group lies in G2G_2 are called G2G_2-manifolds.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Determinant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionNonassociative ring, concept, no evidence levelNonassociative ringconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionThe quaternions, example, no evidence levelThe quaternionsexampleUnitary group, object, no evidence levelUnitary groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectOctonions, object, no evidence levelOctonionsobjectExceptional Lie group G2, example, no evidence levelExceptional Lie group…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Determinant concept
  2. Group concept
  3. Identity matrix object
  4. Lie group concept
  5. Matrix definition
  6. Nonassociative ring concept
  7. Vector space definition
  8. Ring definition
  9. Square matrix definition
  10. Transpose definition
  11. Linear map definition
  12. Unit of a ring definition
  13. Conjugate transpose definition
  14. Orthogonal matrix definition
  15. Multilinear map definition
  16. Division ring definition
  17. Unitary matrix definition
  18. Orthogonal group object
  19. Alternating multilinear map definition
  20. The quaternions example
  21. Unitary group object
  22. Determinant definition
  23. Special unitary group SU(n) example
  24. Special orthogonal group object
  25. Octonions object
  26. Exceptional Lie group G2 example

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/G2_(mathematics)
About this page
TagYX5FFSCR8P
URIm2://YX5FFSCR8P
Kindexample
WritingW0 stub
MSC22E15, 17B25, 22C05
Instance ofSimple Lie group
UsesOctonions, Special orthogonal group, Special unitary group SU(n)

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