What is the definition of the duality group $E_{7(7)}$?

Question

1. What is the definition of the duality group $E_{7(7)}$ that appears in ${\cal N}=8$ Supergravity and what are the basics properties?

2. Moreover what is the relation with the Lie Algebra $E_7$?

3. Please, provide some reference in your answer.

Answer 1

As Wikipedia explains, $E_7$ refers to several, closely related real and complex Lie groups and Lie algebras.

All the various $E_7$ Lie groups (algebras) are Lie subgroups (subalgebras) of the complex Lie group $E_7$ (algebra $e_7$), respectively. The latter has complex dimension $133$ and rank $7$.

Specifically, $E_{7(7)}\equiv E_{7(+7)}\equiv E_{7,7}$ is a real Lie group, where the corresponding Lie algebra is a split Lie algebra, and its Killing form is a split real form. The $E_{7,7}$ Lie group is a subgroup of $Sp(56;\mathbb{R})$.

For the actual group constructions, see e.g. arXiv:1007.4758.

The non-compact Lie group $E_{7,7}$ is the duality group of ${\cal N}=8$ supergravity in $D=4$. The discrete $U$-duality group of type II string theory compactified on a six-torus $T^6$ is $$E_{7,7}(\mathbb{Z}):=E_{7,7} \cap Sp(56;\mathbb{Z}),$$ as originally conjectured in arXiv:hep-th/9410167.