lie correspondence造句

• Apply the Lie correspondence theorem with being the differential at the identity of.
• The Lie correspondence is subsequently employed for obtaining group representations of the connected component of the Lorentz group,.
• Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups.
• Under the Lie correspondence, nilpotent ( respectively, solvable ) connected Lie groups correspond to nilpotent ( respectively, solvable ) Lie algebras.
• The finite-dimensional representations of the connected component of the full Lorentz group are obtained by employing the Lie correspondence and the matrix exponential.
• Motivated by this and the Lie correspondence theorem stated above, let for some vector space be a representation and tentatively define a representation of by first setting
• This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of the Lie correspondence and in quantum field theory.
• As a corollary, since the covering map is a homomorphism, the mapping version of the Lie correspondence can be used to provide a proof of the surjectiveness of for.
• The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.
• The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of space inversion are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.
• By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with ( abelian ) nontrivial non-discrete normal subgroups replaced by ( one-dimensional ) nontrivial ideals in the Lie algebra, and the center of replaced by the center of.
• It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms : one is the "'compact real form "'and corresponds to a compact Lie group under the Lie correspondence ( its Satake diagram has all vertices blackened ), and the other is the "'split real form "'and corresponds to a Lie group that is as far as possible from being compact ( its Satake diagram has no vertices blackened and no arrows ).