“Lie correspondence”的概念、定义、习题解题方法及翻译-数学辞典

单词

Lie correspondence

释义

Lie correspondence

Sophus Lie proved three important theorems about Lie groups and Lie algebras. The third theorem details the correspondence between the two and states that every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group. There is also a correspondence between homomorphisms of algebras and groups and between subalgebras and subgroups.

For example, let G denote the group of n×n matrices with positive determinant. This has Lie algebra Mn×n(R), the space of n×n matrices with Lie bracket [A,B] = AB–BA. The exponential map exp:Mn×n(R) → G, taking a matrix to its exponential, has an important role in Lie's theory. Now the determinant map det: G → (0,∞) is a homomorphism of Lie groups, which must correspond to a homomorphism of the Lie algebras Mn×n(R) → R; in this case that homomorphism is trace. Further, as

is a commutative diagram, this means that

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