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A connected compact complex Lie group A of dimension g is of the form /, a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra a {\displaystyle {\mathfrak {a}}} can be shown to be abelian and then exp : a → A {\displaystyle \operatorname {exp} :{\mathfrak {a}}\to A} is a surjective morphism of complex Lie groups ...
If G is connected with Lie algebra 𝖌, then its universal covering group G is simply connected. Let G C be the simply connected complex Lie group with Lie algebra 𝖌 C = 𝖌 ⊗ C, let Φ: G → G C be the natural homomorphism (the unique morphism such that Φ *: 𝖌 ↪ 𝖌 ⊗ C is the canonical inclusion) and suppose π: G → G is the universal covering map, so that ker π is the ...
The fundamental group of the adjoint form of E 6 (as a complex or compact Lie group) is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. For the simply-connected form, its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface .
An example of a G-structure is an almost complex structure, that is, a reduction of a structure group of an even-dimensional manifold to GL(n,C). Such a reduction is uniquely determined by a C ∞-linear endomorphism J ∈ End(TM) such that J 2 = −1. In this situation, the torsion can be computed explicitly as follows.
One way to define complex tori [1] is as a compact connected complex Lie group.These are Lie groups where the structure maps are holomorphic maps of complex manifolds. It turns out that all such compact connected Lie groups are commutative, and are isomorphic to a quotient of their Lie algebra = whose covering map is the exponential map of a Lie algebra to its associated Lie group.
However, about any point, a generalized complex manifold is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like Weinstein's theorem for the local structure of Poisson manifolds. The remaining question of the local structure is: what does a ...
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group [1] first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method.
The companion concept to associated bundles is the reduction of the structure group of a -bundle . We ask whether there is an H {\displaystyle H} -bundle C {\displaystyle C} , such that the associated G {\displaystyle G} -bundle is B {\displaystyle B} , up to isomorphism .