“Lie correspondence”的概念、定义、翻译、参考文献-科学参考

Lie correspondence

Mathematics

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 M_n×n(ℝ), the space of n×n matrices with Lie bracket [A,B] = AB–BA. The exponential map exp:M_n×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 M_n×n(ℝ) → ℝ; in this case that homomorphism is trace. Further, as

$\begin{matrix} \begin{matrix} M_{n \times n} (ℝ) \\ \begin{matrix} tr & ↓ \end{matrix} \\ ℝ \end{matrix} & \begin{matrix} \overset{exp}{\to} \\ \overset{exp}{\to} \end{matrix} & \begin{matrix} G \\ \begin{matrix} ↓ & det \end{matrix} \\ (0, \infty) \end{matrix} \end{matrix}$

is a commutative diagram, this means that

$det (exp A) = exp (trace A) .$

单词	Lie correspondence
释义	Lie correspondence Mathematics 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 M_n×n(ℝ), the space of n×n matrices with Lie bracket [A,B] = AB–BA. The exponential map exp:M_n×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 M_n×n(ℝ) → ℝ; in this case that homomorphism is trace. Further, as $\begin{matrix} \begin{matrix} M_{n \times n} (ℝ) \\ \begin{matrix} tr & ↓ \end{matrix} \\ ℝ \end{matrix} & \begin{matrix} \overset{exp}{\to} \\ \overset{exp}{\to} \end{matrix} & \begin{matrix} G \\ \begin{matrix} ↓ & det \end{matrix} \\ (0, \infty) \end{matrix} \end{matrix}$ is a commutative diagram, this means that $det (exp A) = exp (trace A) .$
随便看	telemedicine telemetry telemetry system teleo-functionalism teleological argument teleology TeLEOS-1 Teleostei telepathy telephony telephoto lens teleportation telepresence teleprinter telerobotics telescope telescope drive telescope mounting telescopes, optical astronomical telescoping series Telescopio Nazionale Galileo Telescopium teleseism teleshopping Telesio, Bernardino (1509–88)