Math 2.0

Topological groups, Lie groups Classical Lie groups Example

List of simple Lie groups

Builds on Special unitary group SU(n)Spin groupCompact symplectic groupExceptional Lie group G2Exceptional Lie group E8

Example

Every compact connected simple Lie group is isomorphic to a quotient G~/C\tilde{G}/C. Here G~\tilde{G} is exactly one group of the following list, and CC is a subgroup of the centre of G~\tilde{G}. Each G~\tilde{G} is compact, connected and simply connected, and its centre is finite.

Type G~\tilde{G} Dimension Centre
AnA_n, n≥1n \ge 1 SU(n+1)SU(n+1) n(n+2)n(n+2) ℤ/(n+1)\Z/(n+1)
BnB_n, n≥2n \ge 2 Spin(2n+1)\mathrm{Spin}(2n+1) n(2n+1)n(2n+1) ℤ/2\Z/2
CnC_n, n≥3n \ge 3 Sp(n)\mathrm{Sp}(n) n(2n+1)n(2n+1) ℤ/2\Z/2
DnD_n, n≥4n \ge 4 Spin(2n)\mathrm{Spin}(2n) n(2n−1)n(2n-1) ℤ/4\Z/4 for nn odd, ℤ/2×ℤ/2\Z/2 \times \Z/2 for nn even
G2G_2 G2G_2 1414 trivial
F4F_4 F4F_4 5252 trivial
E6E_6 E6E_6 7878 ℤ/3\Z/3
E7E_7 E7E_7 133133 ℤ/2\Z/2
E8E_8 E8E_8 248248 trivial

Here SU(n+1)SU(n+1) is the special unitary group, Spin(m)\mathrm{Spin}(m) is the spin group, and Sp(n)\mathrm{Sp}(n) is the compact symplectic group. The groups G2 and E8 are the smallest and the largest exceptional groups. The type is the type of the Dynkin diagram of the Lie algebra, and nn is the rank. Check of one row: SU(n+1)SU(n+1) has Lie algebra of trace-zero skew-Hermitian (n+1)×(n+1)(n+1) \times (n+1) matrices, of dimension (n+1)2−1=n(n+2)(n+1)^2 - 1 = n(n+2).

Remarks

The bounds on nn avoid repetitions and non-simple cases: Sp(1)≅Spin(3)≅SU(2)\mathrm{Sp}(1) \cong \mathrm{Spin}(3) \cong SU(2), Sp(2)≅Spin(5)\mathrm{Sp}(2) \cong \mathrm{Spin}(5), Spin(6)≅SU(4)\mathrm{Spin}(6) \cong SU(4), and Spin(4)≅SU(2)×SU(2)\mathrm{Spin}(4) \cong SU(2) \times SU(2) is not simple. The first four rows are the classical families, and the last five are the exceptional groups. For example, SO(3)=SU(2)/{±I}SO(3) = SU(2)/\set{\pm I} is the quotient for A1A_1 with CC the whole centre. The same list of types classifies the complex simple Lie algebras. The noncompact simple Lie groups are classified through the real forms of these complex Lie algebras, a classification due to Élie Cartan.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

Show the path as a graph
Clifford algebra, concept, no evidence levelClifford algebraconceptCovering space, concept, no evidence levelCovering spaceconceptCross product, concept, no evidence levelCross productconceptDeterminant, 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 ringconceptReal projective space, object, no evidence levelReal projective spaceobjectSymplectic group, object, no evidence levelSymplectic groupobjectVector 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 transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleOctonions, object, no evidence levelOctonionsobjectExceptional Lie group G2, example, no evidence levelExceptional Lie group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleExceptional Lie group E8, example, no evidence levelExceptional Lie group…exampleSU(2), example, no evidence levelSU(2)exampleSpin group, example, no evidence levelSpin groupexampleCompact symplectic group, example, no evidence levelCompact symplectic gr…exampleList of simple Lie groups, example, no evidence levelList of simple Lie gr…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. Clifford algebra concept
  2. Covering space concept
  3. Cross product concept
  4. Determinant concept
  5. Group concept
  6. Identity matrix object
  7. Lie group concept
  8. Matrix definition
  9. Nonassociative ring concept
  10. Real projective space object
  11. Symplectic group object
  12. Vector space definition
  13. Ring definition
  14. Square matrix definition
  15. Transpose definition
  16. Linear map definition
  17. Unit of a ring definition
  18. Conjugate transpose definition
  19. Multilinear map definition
  20. Orthogonal matrix definition
  21. Division ring definition
  22. Alternating multilinear map definition
  23. Orthogonal group object
  24. Unitary matrix definition
  25. The quaternions example
  26. Determinant definition
  27. Unitary group object
  28. Special orthogonal group object
  29. Special unitary group SU(n) example
  30. Octonions object
  31. Exceptional Lie group G2 example
  32. Rotation group SO(3) example
  33. Exceptional Lie group E8 example
  34. SU(2) example
  35. Spin group example
  36. Compact symplectic group example
  37. List of simple Lie groups example

Further reading

  1. Web https://en.wikipedia.org/wiki/Simple_Lie_group
About this page
Tag2EPRMNK1G2
URIm2://2EPRMNK1G2
Kindexample
WritingW0 stub
MSC22E46, 22C05, 17B25
IllustratesSimple Lie group
UsesSpecial unitary group SU(n), Spin group, Compact symplectic group, Exceptional Lie group G2, Exceptional Lie group E8
Changed2 November 2026

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