18.2 Semi-direct product groups

The Euclidean group example of the previous section can be generalized to the following:

Definition (Semi-direct product group)

Given a group $K,$ a group $N,$ and an action $Φ$ of $K$ on $N$ by automorphisms
$Φ_{k} : n \in N \to Φ_{k} (n) \in N$

the semi-direct product $N \times K$ is the set of pairs $(n, k) \in N \times K$ with group law

(n_{1}, k_{1}) (n_{2}, k_{2}) = (n_{1} Φ_{k_{1}} (n_{2}), k_{1} k_{2})

One can easily check that this satisfies the group axioms. The inverse is

(n, k)^{- 1} = (Φ_{k^{- 1}} (n^{- 1}), k^{- 1})

Checking associativity, one finds

((n_{1}, k_{1}) (n_{2}, k_{2})) (n_{3}, k_{3}) = (n_{1} Φ_{k_{1}} (n_{2}), k_{1} k_{2}) (n_{3}, k_{3}) = (n_{1} Φ_{k_{1}} (n_{2}) Φ_{k_{1} k_{2}} (n_{3}), k_{1} k_{2} k_{3}) = (n_{1} Φ_{k_{1}} (n_{2}) Φ_{k_{1}} (Φ_{k_{2}} (n_{3})), k_{1} k_{2} k_{3}) = (n_{1} Φ_{k_{1}} (n_{2} Φ_{k_{2}} (n_{3})), k_{1} k_{2} k_{3}) = (n_{1}, k_{1}) (n_{2} Φ_{k_{2}} (n_{3}), k_{2} k_{3}) = (n_{1}, k_{1}) ((n_{2}, k_{2}) (n_{3}, k_{3}))

The notation $N \times K$ for this construction has the weakness of not explicitly indicating the automorphism $Φ$ which it depends on. There may be multiple possible choices for $Φ$ , and these will always include the trivial choice $Φ_{k} = 1$ for all $k \in K$ , which will give the standard product of groups.

Digression. For those familiar with the notion of a normal subgroup, $N$ is a normal subgroup of $N \times K$ . A standard notation for $^{66} N$ is a normal subgroup of $G^{s}$ is $N ◃ G$ . The symbol $N \times K$ is supposed to be a mixture of the × and ✁ symbols (note that some authors define it to point in the other direction).

The Euclidean group $E (d)$ is an example with $N = R^{d}, K = S O (d)$ . For $a \in R^{d}, R \in S O (d)$ one has

Φ_{R} (a) = R a

In chapter 42 we will see another important example, the Poincar´e group which generalizes $E (3)$ to include a time dimension, treating space and time according to the principles of special relativity.

The most important example for quantum theory is:

Definition (Jacobi group)

The Jacobi group in $d$ dimensions is the semi-direct product group
$G^{J} (d) = H_{2 d + 1} ⋊ S p (2 d, R)$
If we write elements of the group as
$(((c _{p} c _{q}), c), k)$
where $k \in S p (2 d, R)$ , then the automorphism $Φ_{k}$ that defines the Jacobi group is given by the one studied in section 16.2

$Φ_{k} (((c _{p} c _{q}), c)) = (k (c _{p} c _{q}), c) (18.1)$

Note that the Euclidean group $E (d)$ is a subgroup of the Jacobi group $G^{J} (d)$ the subgroup of elements of the form

(((c _{p} 0), 0), (R 0 0 R))

where $R \in S O (d)$ . The

((c _{p} 0), 0) \subset H_{2 d + 1}

make up the group $R^{d}$ of translations in the $q_{j}$ coordinates, and the

k = (R 0 0 R) \subset S p (2 d, R)

are symplectic transformations since

Ω (k (c _{p} c _{q}), k (c _{p}^{'} c _{q}^{'})) = R c_{q} \cdot R c_{p}^{'} - R c_{p} \cdot R c_{q}^{'} = c_{q} \cdot c_{p}^{'} - c_{p} \cdot c_{q}^{'} = Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'}))

( $R$ is orthogonal so preserves dot products).

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 204、205、206、207、208、209

来源版本：2025-10-20

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18.2 Semi-direct product groups

18.2 Semi-direct product groups

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