Math 2.0

Topological groups, Lie groups Classical Lie groups Example

Exceptional Lie group E8

Builds on Exceptional Lie group G2

Example

The compact Lie group E8E_8 is the compact, connected and simply connected simple Lie group whose Lie algebra has the root system of type E8E_8. The root system E8E_8 lives in ℝ8\R^8 and consists of the 240240 vectors of the following two kinds:

  1. the 112112 vectors with two entries ±1\pm 1 and six entries 00;
  2. the 128128 vectors (±12,…,±12)(\pm \tfrac12, \dots, \pm \tfrac12) with an even number of minus signs.

Every root has squared length 22. Check of the count: there are (82)⋅4=112\binom{8}{2} \cdot 4 = 112 vectors of the first kind and 28/2=1282^8 / 2 = 128 of the second kind.

The dimension of E8E_8 is the rank plus the number of roots, 8+240=2488 + 240 = 248.

Remarks

E8E_8 is the largest of the five exceptional compact simple Lie groups G2,F4,E6,E7,E8G_2, F_4, E_6, E_7, E_8, see G2 for the smallest. Its Dynkin diagram is a tree with 88 nodes and branch lengths 11, 22 and 44 from the branch point. The compact group E8E_8 has trivial centre, so it is at the same time simply connected and adjoint. Its smallest nontrivial representation is the adjoint representation, of dimension 248248. The 240240 roots are the shortest nonzero vectors of the E8 lattice, which is spanned by them. The Weyl group of E8E_8 has order 696 729 600696\,729\,600. The name E8E_8 also denotes the complex simple Lie group of this type and its real forms.

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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…exampleExceptional Lie group E8, 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.
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  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
  27. Exceptional Lie group E8 example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/E8_(mathematics)
About this page
TagMKT1263M0A
URIm2://MKT1263M0A
Kindexample
WritingW0 stub
MSC22E15, 17B25, 22E46
Instance ofSimple Lie group
UsesExceptional Lie group G2
Changed2 November 2026

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