Definition
E7 (complex, simply connected)
E7 (complex, simply connected) as a separately identified complex Lie group, with its defining operations and dimension.
The group is the connected simply connected complex Lie group whose Lie algebra is the complex simple Lie algebra of Dynkin type . The simply connected global form is part of this definition.
Dimensions and structure
This group has real dimension , complex dimension . Its rank is 7. The center is cyclic of order 2. Passing to the adjoint quotient changes which representations descend. The compact, split real, and complex entries are separate objects.
Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.
References
- Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §40.1, p. 213; §42.2, pp. 225–229; §43.1, Theorem 43.4, pp. 230–231.