18.3 Semi-direct product Lie algebras

18.3 Semi-direct product Lie algebras

Concept links · terms present in this machine draft; source roles are unverified: vector space · Lie algebra · Lie group · Lie bracket

We have seen that semi-direct product Lie groups can be constructed by taking a product $N\times K$ of Lie groups as a set, and imposing a group multiplication law that uses an action of $K$ on $N$ by automorphisms. In a similar manner, semi-direct product Lie algebras $\mathfrak{n}⋊\mathfrak{k}$ can be constructed by taking the direct sum of n and k as vector spaces, and defining a Lie bracket that uses an action of $k$ on $n$ by derivations (the infinitesimal version of automorphisms, see equation 16.17).

Considering first the example $E(d)={\mathbf{R}}^d⋊SO(d)$ , recall that elements $E(d)$ can be written in the form

$$\left(\begin{array}{cc}R& \mathbf{a}\\ 0& 1\end{array}\right)$$

for $R\in SO(d)$ and $\mathbf{a}\in {\mathbf{R}}^d$ . The tangent space to this group at the identity will be given by matrices of the form

$$\left(\begin{array}{cc}X& \mathbf{a}\\ 0& 0\end{array}\right)$$

where $X$ is an antisymmetric d by $d$ matrix and $\mathbf{a}\in {\mathbf{R}}^d$ . Exponentiating such matrices will give elements of $E(d)$

The Lie bracket is then given by the matrix commutator

$$\left[\left(\begin{array}{cc}{X}_1& {\mathbf{a}}_1\\ 0& 0\end{array}\right),\left(\begin{array}{cc}{X}_2& {\mathbf{a}}_2\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}[X_1,X_2]& X_1{\mathbf{a}}_2-X_2{\mathbf{a}}_1\\ 0& 0\end{array}\right)\text{\hspace{1em}(18.2)}$$

We see that the Lie algebra of $E(d)$ will be given by taking the sum of ${\mathbf{R}}^d$ (the Lie algebra of ${\mathbf{R}}^d)$ and so $(d)$ , with elements pairs $(\mathbf{a},X)$ with $\mathbf{a}\in {\mathbf{R}}^d$ and $X$ an antisymmetric d by $d$ matrix. The infinitesimal version of the rotation action of $SO(d)$ on ${\mathbf{R}}^d$ by automorphisms

$${\Phi }_R(a)=Ra$$

is

$$\frac{d}{dt}{\Phi }_{e^{tX}}(\mathbf{a}{)}_{|t=0}=\frac{d}{dt}(e^{tX}\mathbf{a}{)}_{|t=0}=X\mathbf{a}$$

Just in terms of such pairs, the Lie bracket can be written

$$[({\mathbf{a}}_1,X_1),({\mathbf{a}}_2,X_2)]=(X_1{\mathbf{a}}_2-X_2{\mathbf{a}}_1,[X_1,X_2])$$

We can define in general:

Definition (Semi-direct product Lie algebra)

Given Lie algebras $k$ and n, and an action of elements $Y\in \mathfrak{k}$ on $n$ by derivations

$$X\in \mathfrak{n}\to Y\cdot X\in \mathfrak{n}$$

the semi-direct product n⋊$k$ is the set of pairs $(X,Y)\in \mathfrak{n}\oplus \mathfrak{k}$ with the Lie bracket

$$[(X_1,Y_1),(X_2,Y_2)]=([X_1,X_2]+Y_1\cdot X_2-Y_2\cdot X_1,[Y_1,Y_2])$$

One can easily see that in the special case of the Lie algebra of $E(d)$ this agrees with the construction above.

In section 16.1.2 we studied the Lie algebra of all polynomials of degree at most two in $d$ dimensional phase space coordinates $q_j,p_j$ , with the Poisson bracket as Lie bracket. There we found two Lie subalgebras, the degree zero and one polynomials (isomorphic to ${\mathfrak{h}}_{2d+1})$ , and the homogeneous degree two polynomials (isomorphic to $\mathfrak{sp}(2d,\mathbf{R}))$ with the second subalgebra acting on the first by derivations as in equation 16.22.

Recall from chapter 16 that elements of this Lie algebra can also be written as pairs

$$\left(\left(\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q}{{\mathbf{c}}_p}\right),c\right),L\right)$$

of elements in ${\mathfrak{h}}_{2d+1}$ and sp(2d, $R$), with this pair corresponding to the polynomial

$${\mu }_L+{\mathbf{c}}_q\cdot \mathbf{q}+{\mathbf{c}}_p\cdot \mathbf{p}+c$$

In terms of such pairs, the Lie bracket is given by

$$\begin{array}{c}\left[\left(\left(\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q}{{\mathbf{c}}_p}\right),c\right),L\right),\left(\left(\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q^{\prime }}{{\mathbf{c}}_p^{\prime }}\right),c\right),L^{\prime }\right)\right]=\\ \left(\left(L\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q^{\prime }}{{\mathbf{c}}_p^{\prime }}\right)-L^{\prime }\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q}{{\mathbf{c}}_p}\right),\Omega \left(\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q}{{\mathbf{c}}_p}\right),\left(\genfrac{}{}{0ex}{}{{\mathbf{c}}_q^{\prime }}{{\mathbf{c}}_p^{\prime }}\right)\right)\right),[L,L^{\prime }]\right)\end{array}$$

which satisfies the definition above and defines the semi-direct product Lie algebra

$${\mathfrak{g}}^J(d)={\mathfrak{h}}_{2d+1}⋊\mathfrak{sp}(2d,\mathbf{R})$$

The fact that this is the Lie algebra of the semi-direct product group

$$G^J(d)=H_{2d+1}⋊Sp(2d,\mathbf{R})$$

follows from the discussion in section 16.2.

The Lie algebra of $E(d)$ will be a sub-Lie algebra of ${\mathfrak{g}}^J(d)$ , consisting of elements of the form

$$\left(\left(\left(\genfrac{}{}{0ex}{}{0}{{\mathbf{c}}_p}\right),0\right),\left(\begin{array}{cc}X& 0\\ 0& X\end{array}\right)\right)$$

where $X$ is an antisymmetric d by $d$ matrix.

Digression. Just as $E(d)$ can be identified with a group of $d$+1 by $d$+1 matrices, the Jacobi group $G_J(d)$ is also a matrix group and one can in principle work with it and its Lie algebra using usual matrix methods. The construction is slightly complicated and represents elements of $G_J(d)$ as matrices in $Sp(2d+1,\mathbf{R})$ . See section 8.5 of [9] for details of the $d=1$ case.

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