5.3 A summary

对照

Before turning to Lie algebra representations, we’ll summarize here the classes of Lie groups and Lie algebras that we have discussed and that we will be studying specific examples of in later chapters:

• The general linear groups $\mathcal{G}\mathcal{L}(n,\mathbf{R})$ and $GL(n,\mathbf{C})$ are the groups of all invertible matrices, with real or complex entries respectively. Their Lie algebras are $\mathfrak{gl}(n,\mathbf{R})=M(n,\mathbf{R})$ and $\mathfrak{gl}(n,\mathbf{C})=M(n,\mathbf{C})$ . These are the vector spaces of all $n$ by $n$ matrices, with Lie bracket the matrix commutator. Other Lie groups will be subgroups of these, with Lie algebras sub-Lie algebras of these Lie algebras.

• The special linear groups $SL(n,\mathbf{R})$ and $SL(n,\mathbf{C})$ are the groups of invertible matrices with determinant one. Their Lie algebras $\mathfrak{sl}(n,\mathbf{R})$ and $\mathfrak{sl}(n,\mathbf{C})$ are the Lie algebras of all $n$ by $n$ matrices with zero trace.

• The orthogonal group $O(n)\subset GL(n,\mathbf{R})$ is the group of $n$ by $n$ real matrices $\Omega$ satisfying ${\Omega }^T={\Omega }^{-1}$. Its Lie algebra $\mathfrak{o}(n)$ is the Lie algebra of $n$ by $n$ real matrices $X$ satisfying $X^T=-X$.

• The special orthogonal group $SO(n)\subset SL(n,\mathbf{R})$ is the subgroup of $O(n)$ with determinant one. It has the same Lie algebra as $O(n):\mathfrak{so}(n)=\mathfrak{o}(n)$

• The unitary group $U(n)\subset GL(n,\mathbf{C})$ is the group of $n$ by $n$ complex matrices $\Omega$ satisfying ${\Omega }^{†}={\Omega }^{-1}$ . Its Lie algebra $\mathfrak{u}(n)$ is the Lie algebra of $n$ by $n$ skew-Hermitian matrices $X$, those satisfying $X^{†}=-X$

• The special unitary group $\mathcal{S}\mathcal{U}(n)\subset \mathcal{SL}(n,\mathbf{C})$ is the subgroup of $U(n)$ of matrices of determinant one. Its Lie algebra $\mathfrak{su}(n)$ is the Lie algebra of $n$ by $n$ skew-Hermitian matrices $X$ with trace zero.

In later chapters we’ll encounter some other examples of matrix Lie groups, including the symplectic group $Sp(2d,\mathbf{R})$ (see chapter 16) and the pseudoorthogonal groups $O(r,s)$ (see chapter 29).

---

上一节：5.2 Lie algebras of the orthogonal and unitary groups · 下一节：5.4 Lie algebra representations

附录：约定与记号

来源与版本

正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 53、54

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837