16.2 The symplectic group and automorphisms of the Heisenberg group

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Returning to the $d = 1$ case, we have found two three dimensional Lie algebras $(h_{3}$ and $sl (2, R))$ ) as subalgebras of the infinite dimensional Lie algebra of functions on phase space:

• $h_{3}$ , the Lie algebra of linear polynomials on $M,$ with basis 1, $q, p .$

$sl (2, R)$ , the Lie algebra of order two homogeneous polynomials on $M .$ with basis $q^{2}, p^{2}, q p$

Taking all quadratic polynomials, we get a six dimensional Lie algebra with basis elements $1, q, p, q p, q^{2}, p^{2}$ . This is not the direct product of $h_{3}$ and $sl (2, R)$ since there are nonzero Poisson brackets

{q p, q} {\frac{p ^{2}}{2}, q} = - q, {q p, p} = p = - p, {\frac{q ^{2}}{2}, p} = q (16.15)

These relations show that operating on a basis of linear functions on $M$ by taking the Poisson bracket with something in $sl (2, R)$ (a quadratic function) provides a linear transformation on $M^{*} = M$

In this section we will see that this is the infinitesimal version of the fact that $S L (2, R)$ ) acts on the Heisenberg group $H_{3}$ by automorphisms. We’ll begin with a general discussion of what happens when a Lie group $G$ acts by automorphisms on a Lie group $H$ , then turn to two examples: the conjugation action of $G$ on itself and the action of $S L (2, R)$ on $H_{3}$

An action of one group on another by automorphisms means the following:

Definition (Group automorphisms)

If an action of elements $g$ of a group $G$ on a group $H$
$h \in H \to Φ_{g} (h) \in H$

satisfies

Φ_{g} (h_{1}) Φ_{g} (h_{2}) = Φ_{g} (h_{1} h_{2})

for all $g \in G$ and $h_{1}, h_{2} \in H$ , the group $G$ is said to act on $H$ by automorphisms. Each map $Φ_{g}$ is an automorphism of H. Note that since $Φ_{g}$ is an action of G, we have $Φ_{g_{1} g_{2}} = Φ_{g_{1}} Φ_{g_{2}}$

When the groups are Lie groups, taking the derivative $ϕ_{g} : h \to h$ of the map $Φ_{g} : H H$ at the identity of H gives a Lie algebra automorphism, defined by

Definition (Lie algebra automorphisms)

If an action of elements $g$ of a group $G$ on a Lie algebra $h$
$X \in h \to ϕ_{g} (X) \in h$

satisfies

[ϕ_{g} (X), ϕ_{g} (Y)] = ϕ_{g} ([X, Y])

for all $g \in G$ and $X, Y \in h$ , the group is said to act on $h$ by automorphisms.

Given an action $ϕ_{g}$ of a Lie group $G$ on $h$ , we get an action of elements $Z \in g$ on $h$ by linear maps:

X \to Z \cdot X = \frac{d}{d t} (ϕ_{e^{t} Z} (X))_{∣ t = 0} (16.16)

that we will often refer to as the infinitesimal version of the action $ϕ_{g}$ of $G$ on $h$ . These maps satisfy

Z \cdot [X, Y] = \frac{d}{d t} (ϕ_{e^{tZ}} ([X, Y]))_{∣ t = 0} = \frac{d}{d t} ([ϕ_{e^{tZ}} (X), ϕ_{e^{tZ}} (Y)])_{∣ t = 0} = [Z \cdot X, Y] + [X, Z \cdot Y]

and one can define

Definition (Lie algebra derivations)

If an action of a Lie algebra $g$ on a Lie algebra $h$ by linear maps
$X \in h \to Z \cdot X \in h$

satisfies

[Z \cdot X, Y] + [X, Z \cdot Y] = Z \cdot [X, Y] (16.17)

for all $Z \in g$ and $X, Y \in h$ , the Lie algebra $g$ is said to act on $h$ by derivations. The action of an element $Z$ on $h$ is a derivation of h.

16.2.1 The adjoint representation and inner automorphisms

Concept links · terms present in this machine draft; source roles are unverified: linear map · Lie algebra · adjoint representation · Lie group

Any group $G$ acts on itself by conjugation, with

Φ_{g} (g^{'}) = g g^{'} g^{- 1}

giving an action by automorphisms (these are called “inner automorphisms”). The derivative at the identity of the map $Φ_{g}$ is the linear map on $g$ given by the adjoint representation operators $A d (g)$ discussed in chapter 5. $So,$ in this case the corresponding action by automorphisms on the Lie algebra $g$ is the adjoint action

X \in g \to ϕ_{g} (X) = A d (g) (X) = g X g^{- 1}

The infinitesimal version of the Lie group adjoint representation by $A d (g)$ on $g$ is the Lie algebra adjoint representation by operators $a d (Z)$ on $g$

X \in g \to Z \cdot X = a d (Z) (X) = [Z, X]

This is an action of $g$ on itself by derivations.

16.2.2 The symplectic group as automorphism group

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Recall the definition 13.2 of the Heisenberg group $H_{3}$ as elements

((y x), z) \in R^{2} \oplus R

with the group law

((y x), z) ((y ^{'} x ^{'}), z^{'}) = ((y + y ^{'} x + x ^{'}), z + z^{'} + \frac{1}{2} Ω ((y x), (y ^{'} x ^{'})))

Elements $g \in S L (2, R)$ act on $H_{3}$ by

((y x), z) \to Φ_{g} (((y x), z)) = (g (y x), z) (16.18)

Here $G = S L (2, R), H = H_{3}$ and $Φ_{g}$ given above is an action by automorphisms since

Φ_{g} ((y x), z) Φ_{g} ((y ^{'} x ^{'}), z^{'}) = (g (y x), z) (g (y ^{'} x ^{'}), z^{'}) = (g (y + y ^{'} x + x ^{'}), z + z^{'} + \frac{1}{2} Ω (g (y x), g (y ^{'} x ^{'}))) = (g (y + y ^{'} x + x ^{'}), z + z^{'} + \frac{1}{2} Ω ((y x), (y ^{'} x ^{'}))) = Φ_{g} (((y x), z) ((y ^{'} x ^{'}), z^{'})) (16.19)

Recall that, in the exponential coordinates we use, the exponential map between the Lie algebra $h_{3}$ and the Lie group $H_{3}$ is the identity map, with both h₃ and $H_{3}$ identified with $R^{2} \oplus R$ . As in section 14.2. we will explicitly identify $h_{3}$ with functions $c_{q} q + c_{p} p + c$ on $M$ , writing these as

((c _{p} c _{q}), c)

with Lie bracket

[((c _{p} c _{q}), c), ((c _{p}^{'} c _{q}^{'}), c^{'})] = ((0 0), c_{q} c_{p}^{'} - c_{p} c_{q}^{'}) = ((0 0), Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'})))

The linearized action of $Φ_{g}$ at the identity of $H_{3}$ gives the action $ϕ_{g}$ on $h_{3} .$ but since the exponential map is the identity, $ϕ_{g}$ acts on $R^{2} \oplus R = M \oplus R$ in the same way as $Φ_{g}$ , by

((c _{p} c _{q}), c) \in h_{3} \to ϕ_{g} (((c _{p} c _{q}), c)) = (g (c _{p} c _{q}), c)

Since the Lie bracket just depends on Ω, which is $S L (2, R)$ invariant, $ϕ_{g}$ preserves the Lie bracket and so acts by automorphisms on $h_{3}$ .

The infinitesimal version of the $S L (2, R)$ action $ϕ_{g}$ on $h_{3}$ is an action of $sl (2, R)$ on $h_{3}$ by derivations. This action can be found by computing (for $L \in sl (2, R)$ and $X \in h_{3})$ using equation 16.16 to get

L \cdot ((c _{p} c _{q}), c) = \frac{d}{d t} (ϕ_{e^{t L}} ((c _{p} c _{q}), c))_{∣ t = 0} = (L (c _{p} c _{q}), 0) (16.20)

The Poisson brackets between degree two and degree one polynomials discussed at the beginning of this section give an alternate way of calculating this action of $sl (2, R)$ on $h_{3}$ by derivations. For a general $L \in sl (2, R)$ (see equation 16.6) and $c_{q} q + c_{p} p + C \in h_{3}$ we have

{μ_{L}, c_{q} q + c_{p} p + C} = c_{q}^{'} q + c_{p}^{'} p, (c _{p}^{'} c _{q}^{'}) = (c c _{q} - a c _{p} a c _{q} + b c _{p}) = L (c _{p} c _{q}) (16.21)

(here $μ_{L}$ is given by 16.9). We see that this is the action of sl(2, $R$ ) by derivations on $h_{3}$ of equation 16.20, the infinitesimal version of the action of $S L (2, R)$ on $h_{3}$ by automorphisms.

Note that in the larger Lie algebra of all polynomials on $M$ of order two or less, the action of $sl (2, R)$ on $h_{3}$ by derivations is part of the adjoint action of the Lie algebra on itself, since it is given by the Poisson bracket (which is the Lie bracket), between order two and order one polynomials.

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16.2 The symplectic group and automorphisms of the Heisenberg group