Group Theory

\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



One of the exceptional Lie groups.


As automorphisms of the exceptional Jordan algebra

The group of determinant-preserving linear isomorphisms of the vector space underlying the octonionic Albert algebra is E 6(26){E}_{6(-26)}. (see e.g. (Manogue-Dray 09)).

This may be written as SL(3,𝕆)SL(3,\mathbb{O}). (Dray-Manogue 09a (16))

graphics grabbed from (Dray-Manogue 09a, p. 12)

As part of the ADE pattern

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

As U-duality group of5d SuGra

E 6E_6 is the U-duality group (see there) of 11-dimensional supergravity compactified to 5 dimensions.

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
SL(2,)SL(2,\mathbb{R})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
SU(5)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
Spin(10)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()E_{11(11)}(\mathbb{Z})

(Hull-Townsend 94, table 1, table 2)


A description of E 6(26)E_{6(-26)} as the special linear group in dimension 3 with values in the octonions is given in

  • Corinne Manogue, Tevian Dray, Octonions and the Structure of E 6E_6 Comment. Math. Univ. Carolin., 51:193–207, 2010.

  • Aaron Wangberg, Tevian Dray, E 6E_6, the Group: The structure of SL(3,𝕆)SL(3,\mathbb{O}) (arXiv:1212.3182)

Cohomological properties are discussed in

  • Mamoru Mimura, Yuriko Sambe, Michishige Tezuka, Cohomology mod 3 of the classifying space of the exceptional Lie group E 6E_6, I : structure of Cotor (arXiv:1112.5811),

    Cohomology mod 3 of the classifying space of the exceptional Lie group E 6E_6, II : The Weyl group invariants (arXiv:1201.3414)

See also

Revised on February 1, 2017 03:33:42 by Urs Schreiber (