nLab
sl(2)

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The Lie algebra 𝔰𝔩(2,k)\mathfrak{sl}(2,k) (over some ground field kk) is the special linear Lie algebra 𝔰𝔩(n,k)\mathfrak{sl}(n,k) for n=2n = 2.

Over ground field k=k = \mathbb{R} (real numbers) or k=k = \mathbb{C} (complex numbers) this is the Lie algebra corresponding to the Lie group which is the special linear group SL(2,)SL(2,\mathbb{R}) or SL(2,)SL(2,\mathbb{C}), respectively.

Properties

As the complexification of 𝔰𝔲(2)\mathfrak{su}(2)

𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{C}) is the complexification of the special unitary Lie algebra 𝔰𝔲(2)\mathfrak{su}(2) (see at su(2)) (…)

Jacobson-Morozov theorem

See at Jacobson-Morozov theorem.

Created on December 4, 2019 at 09:26:51. See the history of this page for a list of all contributions to it.