nLab E₇

Idea

One of the exceptional Lie groups.

Definition

Consider the vector space

$$W \coloneqq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast$$

of dimension $56$. This is naturally a symplectic vector space with symplectic form $\omega$ given by the natural pairing between linear 2-forms? and bivectors.

In addition, consider on this space the quartic form $q \colon W \to \mathbb{R}$ which sends an element $v = (\{v^{a b}, w_{a b}\}) \in W$ to

$$q(v) \coloneqq v^{a b } w_{b c} v^{c d} w_{d a} - \tfrac{1}{4} v^{a b} w_{a b} v^{c d} w_{c d} + \tfrac{1}{96} \left( \epsilon_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} v^{a_1 a_2} v^{a_3 a_4} v^{a_5 a_6} v^{a_7 a_8} + \epsilon^{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} w_{a_1 a_2} w_{a_3 a_4} w_{a_5 a_6} w_{a_7 a_8} \right) \,.$$

Now $E_{7(7)} \subset GL(56,\mathbb{R})$ is the subgroup of the general linear group acting on $W$ which preserves both the symplectic form $\omega$ as well as the quartic form $q$. See also below.

This presentation is due to Cartan, for review see Cremmer-Julia 79, appendix B, Pacheco-Waldram 08, B.1. A construction via octonions is due to (Freudenthal 54), one via quaternions is due to (Wilson 2014).

Properties

Representation

$\mathbf{56}$ – The smallest fundamental representation

The smallest fundamental representation of $E_7$ is the defining one (from the definition above), of dimension $56$. Under the special linear subgroup $SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes as (e.g. Cacciatori et al. 10, section 4, also Pacheco-Waldram 08, appendix B)

$$\mathbf{56} \simeq \mathbf{28} \oplus \mathbf{28}^\ast \simeq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast \,.$$

Under the further subgroup inclusion $SL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes further as

$$\mathbf{56} \simeq \underset{\simeq \wedge^2 \mathbb{R}^8}{\underbrace{\mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast}} \oplus \underset{\simeq \wedge^2 (\mathbb{R}^8)^\ast}{\underbrace{\wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7}} \,,$$

where $\wedge^2 (\mathbb{R}^7) \subset \wedge^2 (\mathbb{R}^8)^\ast$ is regarded as the subspace of 2-forms with vanishing 8-components, and where $\wedge^6 \mathbb{R}^7$ is the Poincaré dual to the complementary subspace of $\wedge^2 (\mathbb{R}^8^\ast)$ of 2-forms with non-trivial 8-component.

This is due to Cartan, for review see Cremmer-Julia 79, appendix B, Pacheco-Waldram 08, B.1.

$\mathbf{133}$ – The adjoint representation

The adjoint representation $\mathbf{133}$ of $E_7$ decomposes under $SL(8,\mathbb{R})$ as (Pacheco-Waldram 08 (B.7))

$$\mathfrak{e}_7 = \mathbf{133} \simeq (\mathbb{R}^8 \otimes (\mathbb{R}^8)^\ast)_{traceless} \oplus \wedge^4 (\mathbb{R}^8)^\ast \,.$$

In this decomposition the subspace corresponding to the subalgebra $\mathfrak{su}(8) \hookrightarrow \mathfrak{e}_8$ is the vector space

$$\mathfrak{su}(8) \simeq \mathfrak{so}(8) \oplus (\wedge^4 (\mathbb{R}^8)^\ast)_- \,,$$

where the first summand denotes the skew-symmetric matrices, and the second summand the Hodge anti-self dual 4-forms (Pacheco-Waldram 08 (B.29) (B.30) and below (2.34)).

Under $GL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R})$ the full adjoint representation decomposes further into (Pacheco-Waldram 08 (B.21))

$$\mathbf{133} \simeq \left(\mathbb{R}^7 \otimes (\mathbb{R}^7)^\ast\right) \oplus \left(\wedge^6 \mathbb{R}^7 \oplus \wedge^6 (\mathbb{R}^7)^\ast\right) \oplus \left( \wedge^3 \mathbb{R}^7 \oplus \wedge^3 (\mathbb{R}^7)^\ast \right) \,.$$

Here $\wedge^6 (\mathbb{R}^7)^\ast \simeq \mathbb{R}^7$ is the $(-,8)$-component of $\mathbb{R}^7 \oplus (\mathbb{R}^7)^\ast$ and dually, while the $(8,8)$-component carries no information by tracelessness; and $\wedge^3 (\mathbb{R}^7)^\ast$ is the $(-,-,-,8)$-component of $\wedge^4 (\mathbb{R}^8)^\ast$, while $\wedge^3 \mathbb{R}^7$ is the 7-dimensional Poincaré dual of the complement of the $(-,-,-,8)$-component (Pacheco-Waldram 08 (B.22)).

Taken together this means that under $GL(7,\mathbb{R})$ the subspace $\mathbb{su}(8) \hookrightarrow \mathfrak{e}_8$ is that spanned by

1. $\mathfrak{so}(7)$-elements;

2. sums of a 3-form with its 8d-Hodge+7d-Poincaré-dual 3-vector;

3. sums of a 6-form with its dual 6-vector

hence is

$$\mathfrak{su}(8) \simeq \mathfrak{so}(8) \oplus \wedge^3 \mathbb{R}^7 \oplus \wedge^6 \mathbb{R}^7 \,.$$

Hence the tangent space to the coset $E_{7(7)}/(SU(8)/\mathbb{Z}_2)$ may be identified as

$$\mathfrak{e}_7/\mathfrak{su}(8) \simeq \odot^2 (\mathbb{R}^7)^\ast \oplus \wedge^3 (\mathbb{R}^7)^\ast \oplus \wedge^6 (\mathbb{R}^7)^\ast \,.$$

As part of the ADE pattern

ADE classification and McKay correspondence

Dynkin diagram/ Dynkin quiver	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral group $T$	binary tetrahedral group $2T$	E6
$E_7$	cube, octahedron	octahedral group $O$	binary octahedral group $2O$	E7
$E_8$	dodecahedron, icosahedron	icosahedral group $I$	binary icosahedral group $2I$	E8

As U-Duality group of 4d SuGra

$E_{7(7)}$ is the U-duality group (see there) of 11-dimensional supergravity compactified on a 7-dimensional fiber to 4-dimensional supergravity (e.g. M-theory on G2-manifolds).

Specifically, (Hull 07, section 4.4, Pacheco-Waldram 08, section 2.2) identifies the vector space underlying the $SL(7,\mathbb{R})$-decomposition of the smallest fundamental representation

$$\mathbf{56} \simeq \mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast \oplus \wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7 \,.$$

as the exceptional tangent bundle-structure to the 7-dimensional fiber space which one obtains as discussed at M-theory supersymmetry algebra – As an 11-dimensional boundary condition. Here $\mathbb{R}^7$ is the ordinary tangent space itself, $\wedge^2 (\mathbb{R}^\ast)^7$ is interpreted as the local incarnation of the possible M2-brane charges, $\wedge^5 (\mathbb{R}^\ast)^7$ the M5-brane charges and $\wedge^6 \mathbb{R}^7$ as the charges of Kaluza-Klein monopoles.

	supergravity gauge group (split real form)	T-duality group (via toroidal KK-compactification)	U-duality	maximal gauged supergravity
	$SL(2,\mathbb{R})$	1	$SL(2,\mathbb{Z})$ S-duality	D=10 type IIB supergravity
	SL$(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3)$\times$ SU(2)	SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 supergravity	E₈-equivariant elliptic cohomology
E₁₀	$E_{10(10)}$	$O(9,9;\mathbb{Z})$	$E_{10(10)}(\mathbb{Z})$
E₁₁	$E_{11(11)}$	$O(10,10;\mathbb{Z})$	$E_{11(11)}(\mathbb{Z})$

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

Related concepts

• exceptional geometry, exceptional generalized geometry

• G₂, F₄,

  E₆, E₇, E₈, E₉, E₁₀, E₁₁, $\cdots$

References

General

The description of the defining fundamental $\mathbf{56}$-representation of $E_{7(7)}$ is due to

• Eli Cartan, Thesis, in Oeuvres complètes T1, Part I, Gauthier-Villars, Paris 1952

and recalled for instance in

• Eugene Cremmer, Bernard Julia, appendix B of The $SO(8)$ Supergravity, Nucl. Phys. B 159 (1979) 141 (spire)

See also

• Robert B. Brown, Groups of type $E _7$, Jour. Reine Angew. Math. 236 (1969), 79-102.

A construction via the octonions is due to

• Hans Freudenthal, Beziehungen der $\mathfrak{e}_7$ und $\mathfrak{e}_8$ zur Oktavenebene, I, II, Indag. Math. 16 (1954), 218–230, 363–368. III, IV, Indag. Math. 17 (1955), 151–157, 277–285. V — IX, Indag. Math. 21 (1959), 165–201, 447–474. X, XI, Indag. Math. 25 (1963) 457–487 (dspace)

reviewed in

• John Baez, section 4.5 E7 of The Octonions (arXiv:math/0105155)

• Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti, E7 groups from octonionic magic square (arXiv:1007.4758)

A quaternionic construction is given in

• Robert Wilson, A quaternionic approach to $E_7$, Proc. Amer. Math. Soc. 142 (2014), 867-880. doi:10.1090/S0002-9939-2013-11838-1, (pre-publication version), (talk notes).

See also

• Wikipedia, E7

On the “intermediate” $E_{7 \tfrac{1}{2}}$:

• J. M. Landsberg, L. Manivel, The sextonions and $E_{7\tfrac{1}{2}}$, Advances in Mathematics 201 1 (2006) 143-179 [arXiv:math/0402157, doi:10.1016/j.aim.2005.02.001]

• Kimyeong Lee, Kaiwen Sun, Haowu Wang, On intermediate Lie algebra $E_{7 \tfrac{1}{2}}$ [arXiv:2306.09230]

See also:

• Tevian Dray, Corinne Manogue, Robert Wilson, A New Division Algebra Representation of $E_7$ [arxiv:2401.10534]

In view of U-duality

The hidden E7-U-duality symmetry of the KK-compactification of 11-dimensional supergravity on a 7-dimensional fiber to 4d supergravity was first noticed in (Cremmer-Julia 79) and then expanded on in

• Bernard de Wit, Hermann Nicolai, D = 11 Supergravity With Local SU(8) Invariance, Nucl. Phys. B 274, 363 (1986) (spire), Local SU(8) invariance in $d = 11$ supergravity (spire)

The proposal to make this hidden $E_7$-symmetry manifest via exceptional generalized geometry is due to

• Chris Hull, Generalised Geometry for M-Theory, JHEP 0707:079 (2007) (arXiv:hep-th/0701203)

• Paulo Pires Pacheco, Daniel Waldram, appendix B of M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)

Further discussion includes

• Mariana Graña, Jan Louis, Aaron Sim, Daniel Waldram, section 3.1 of $E_{7(7)}$ formulation of $N=2$ backgrounds (arXiv:0904.2333)

On $E_7$-exceptional field theory formulation of D=11 supergravity:

• Olaf Hohm, Henning Samtleben: Exceptional Field Theory II: $E_{7(7)}$, Phys. Rev. D 89 066017 (2014) [arXiv:1312.4542, doi:10.1103/PhysRevD.89.066017]