Seminar Sophus Lie 2 (1991) 31–38

A note on the oscillator group

Norbert Dörr^{∗}

Abstract. In this note we give an alternative parametrization of the oscillator algebra. We determine the semigroup of the oscillator group that is infinitesimally generated by an invariant cone whose exponential image is not dense in the semigroup.

1. An alternative parametrization for the oscillator algebra

Let M(2) be the universal covering group of the group of Euclidian motions of the plane. Then M(2) may be realized as the group of 3 × 3-matrices

( \begin{matrix}e^{ir} & c & 0 \\ 0 & e^{−ir} & 0 \\ 0 & 0 & e^{r} \end{matrix} ) , r ∈ ℝ, c ∈ ℂ.

If we write (c, r) for these matrices, the multiplication is given by

(c, r)·(c′, r′) = (e^{ir}c′ + e^{−ir′}c, r + r′).  (1)

The corresponding Lie algebra 𝔪(2) may be identified with ℂ⊕ℝ with Lie bracket

[(c, r), (c′, s′)] = (2i· det \binom{r r′}{c c′}, 0).  (2)

We prefer this parametrization of M(2) because all one-parameter groups are “flat”, i. e., they are contained in a two-dimensional subspace.

Lemma 1.1. Let M(2) be the group of Euclidian motions of the plane with multiplication (1). Then the one-parameter groups are given by

φ(t) = (ρ· sin r, r),  (3)

where ρ ∈ ℂ.

Proof. This may be verified by direct computation.

∗The author thanks Prof. Dr. Karl H. Hofmann for his support.

Let H be the Heisenberg group. It can be realized as ℝ × ℂ with multiplication

(z, c)·(z′, c′) = (z + z′ + \frac{1}{2} Im(\overline{c}c′), c′ + c),  (4)

where z, z′ ∈ ℝ, c, c′ ∈ ℂ. The Heisenberg algebra 𝔥 = L(H) = ℝ ⊕ ℂ has Lie bracket

[(z, c), (z′, c′)] = (Im(\overline{c}c′), 0)  (5)

for z, z′ ∈ ℝ, c, c′ ∈ ℂ.

We consider the four-dimensional oscillator group O_{4} with corresponding Lie algebra 𝔬_{4} = L(O_{4}). The oscillator group is the semidirect product of ℝ and the Heisenberg group H = ℝ × ℂ, where ℝ acts by rotation on ℂ. Motivated by the parametrization of M(2) we can realize O_{4} in the following way.

Lemma 1.2. The oscillator group O_{4} admits the following parametrization: O_{4} = ℝ × ℂ × ℝ with multiplication

(z, c, r)·(z′, c′, r′) = (z + z′ + \frac{1}{2} Im(e^{i(r+r′)}\overline{c}c′), e^{ir}c′ + e^{−ir′}c, r + r′)  (6)

for z, z′, r, r′ ∈ ℝ, c, c′ ∈ ℂ.

The Lie algebra 𝔬_{4} = L(O_{4}) = ℝ ⊕ ℂ ⊕ ℝ has Lie bracket

[(z, c, r), (z′, c′, r′)] = (Im(\overline{c}c′), 2i· det \binom{r r′}{c c′}, 0)  (7)

for z, z′, r, r′ ∈ ℝ, c, c′ ∈ ℂ.

In what follows, we explicitly compute the exponential function

exp: 𝔬_{4} → O_{4}

of the oscillator group in this parametrization. The shape of the one-parameter-groups in the motion group M(2) motivates the following definition. Set

Φ(t) =^{def} (φ(t), c_{0}· sin t, t),

where c_{0} ∈ ℂ. By a straightforward computation using (6), we get

(φ(t), c_{0}· sin t, t)·(φ(u), c_{0}· sin u, u) =

(φ(t) + φ(u) + \frac{|c_{0}|^{2}}{4} (sin^{2} t sin(2u) + sin(2t) sin^{2} u), c_{0}· sin(t + u), t + u).

Thus we get the following functional equation for φ:

φ(t + u) = φ(t) + φ(u) + \frac{|c_{0}|^{2}}{4} (sin^{2} t sin(2u) + sin(2t) sin^{2} u)

for all t, u ∈ ℝ. If we devide both sides by u and let u tend to zero, we get the differential equation

φ′(t) = φ′(0) + \frac{|c_{0}|^{2}}{2} sin^{2} t

which integrates to

φ(t) = (φ′(0) + \frac{|c_{0}|^{2}}{4} )·t − \frac{|c_{0}|^{2}}{8} sin(2t).

Thus we have

Φ(t) = ((z_{0} + \frac{|c_{0}|^{2}}{4} )·t − \frac{|c_{0}|^{2}}{8} sin(2t), c_{0} sin t, t),  (8)

where z_{0} ∈ ℝ, c_{0} ∈ ℂ, and Φ′(0) = (z_{0}, c_{0}, 1). With these preparations we get the following result.

Proposition 1.3. The exponential function exp: 𝔬_{4} → O_{4} of the oscillator group is given by

exp(z, c, r) = \{ \begin{matrix}(z + \frac{|c|^{2}}{4r} − \frac{|c|^{2}}{8r^{2}} sin(2r), c·\frac{sin r}{r}, r) & for r ≠ 0; \\ (z, c, 0) & for r = 0. \end{matrix}

Proof. This follows from (8) where we have only to take care to get the right parameters. It is also sufficient to note that \frac{d}{dt}|_{t=0} exp t·(z, c, r) = (z, c, r).

2. Invariant wedges in the oscillator algebra

We consider on 𝔬_{4} the symmetric, bilinear form 𝔮: 𝔬_{4} × 𝔬_{4} → ℝ defined by

q((z, c, r), (z′, c′, r′)) =^{def} rz′ + r′z + \frac{1}{2} Re(\overline{c}c′)  (9)

for (z, c, r), (z′, c′, r′) ∈ 𝔬_{4}. The associated quadratic form is denoted by the same letter, i. e.,

q(z, c, r) =^{def} 2rz + \frac{1}{2}|c|^{2}.  (10)

Lemma 2.1. The symmetric, bilinear form 𝔮: 𝔬_{4} × 𝔬_{4} → ℝ defined by (9) is invariant, i. e., we have q([w_{1}, w_{2}], w_{3}) = q(w_{1}, [w_{2}, w_{3}]) for w_{j} ∈ 𝔬_{4}, j = 1, 2, 3.

Proof. Let w_{j} = (z_{j}, c_{j}, r_{j}) ∈ 𝔬_{4}, j = 1, 2, 3. Then we have

q([w_{1}, w_{2}], w_{3}) = q((Im(\overline{c_{1}}c_{2}), 2i· det \binom{r_{1} r_{2}}{c_{1} c_{2}}, 0), (z_{3}, c_{3}, r_{3}))
= s_{3}· Im(\overline{c_{1}}c_{2}) + \frac{1}{2} Re(\overline{2i· det \binom{r_{1} r_{2}}{c_{1} c_{2}}}c_{3})
= s_{3}· Im(\overline{c_{1}}c_{2}) + s_{1}· Im(\overline{c_{2}}c_{3}) − s_{2}· Im(\overline{c_{1}}c_{3})

and

q(w_{1}, [w_{2}, w_{3}]) = q((z_{1}, c_{1}, r_{1}), (Im(\overline{c_{2}}c_{3}), 2i· det \binom{r_{2} r_{3}}{c_{2} c_{3}}, 0))
= s_{1}· Im(\overline{c_{2}}c_{3}) + \frac{1}{2} Re(2i·\overline{c_{1}} det \binom{r_{2} r_{3}}{c_{2} c_{3}})
= s_{1}· Im(\overline{c_{2}}c_{3}) + s_{3}· Im(\overline{c_{1}}c_{2}) − s_{2}· Im(\overline{c_{1}}c_{3})

These two expressions are equal, as required.

Definition 2.2. Denote by W the invariant Lorentz cone defined by the zeroset of the invariant quadratic form q, i. e.,

W =^{def} {(r, v, z) ∈ L | q(z, c, r) ≤ 0, r ≥ 0, z ≤ 0}.  (11)

In what follows, we denote the closed semigroup that is generated by W with

S =^{def} \overline{⟨exp W ⟩}.  (12)

In the next paragraph we show that S is an infinitesimally generated semigroup with tangent wedge L(S) = W. But first, we determine the exponential image exp W of W. We have

(z_{0}, c_{0}, 1) ∈ W ⟺ z_{0} + \frac{1}{4}|c_{0}|^{2} ≤ 0.  (13)

We denote by Φ_{z_{0},c_{0}}: ℝ → O_{4} the one-parameter group generated by (z_{0}, c_{0}, 1). Hence we have by Proposition 1.3

Φ_{z_{0},c_{0}} =^{def} exp t·(z_{0}, c_{0}, 1)
= ((z_{0} + \frac{|c_{0}|^{2}}{4} )·t − \frac{|c_{0}|^{2}}{8} sin(2t), c_{0} sin t, t).

Since (z_{0}, c_{0}, 1) + ℝ^{−}·(1, 0, 0) ⊆ W whenever (z_{0}, c_{0}, 1) ∈ W, the boundary of exp W is just the image of ∂W. Hence we have as “boundary curves” Φ_{c_{0}} of exp W

Φ_{c_{0}}(t) = (−\frac{|c_{0}|^{2}}{8} · sin(2t), c_{0} sin t, t), t ≥ 0,  (14)

i. e., the one-parameter semigroups Φ_{z_{0},c_{0}} with z_{0} = −\frac{|c_{0}|^{2}}{4} .

In Figure 1 we see the projection of Φ_{c_{0}} into ℝ × 0 × ℝ, resp., 0 × ℝ·c_{0} × ℝ.

Figure 1: Projection of Φ_{c_{0}} into ℝ × 0 × ℝ, resp., 0 × ℝ·c_{0} × ℝ.

Lemma 2.3. For the intersection of the exponential image of W with the hyperplane H_{t} =^{def} ℝ × ℂ × {t} the following holds.

(i) For t = k·π, k ∈ ℕ:

(exp W) ∩ H_{t} = ℝ^{−} × {0} × {k·π}.

(ii) For t ≠ k·π, k ∈ ℕ:

(exp W) ∩ H_{t} = \{(−\frac{sin(2t)}{4(1 − cos(2t))}·|c̃|^{2}, c̃, t) | c̃ ∈ ℂ\} + ℝ^{−} × \{0\} × \{0\}.

Proof. The case t = 0 is obvious, since exp(z, c, 0) = (z, c, 0), and (z, c, 0) ∈ W if and only if z ≤ 0 and c = 0.

By Proposition 1.3, we have for t = k·π, k ∈ ℕ \ {0},

exp(z, c, k·π) = (z + \frac{|c|^{2}}{4k·π} , 0, k·π).

Again, the fact, that (z, c, k·π) ∈ W if and only if z+ \frac{|c|^{2}}{4k·π} ≤ 0, proves the assertion in the case t = k·π.

For (z, c, t) ∈ W we have

exp(z, c, t) = (z + \frac{|c|^{2}}{4t} − \frac{|c|^{2}}{8t} sin(2t), c·\frac{sin t}{t}, t),

where z + \frac{|c|^{2}}{4t} ≤ 0. Thus we have

(exp W) ∩ H_{t} = \{(−\frac{|c|^{2}}{8t} sin(2t), c·\frac{sin t}{t}, t) | c ∈ ℂ\} + ℝ^{−} × \{0\} × \{0\}.

Set c̃ = c·\frac{sin t}{t}. Then a straightforward calculation using sin^{2} t = \frac{1}{2}(1 − cos(2t)) proves the assertion.

For a fixed c̃ ∈ ℂ we identify the two-dimensional subspace H_{t,_{c}} =^{def} ℝ ×

ℝ·c̃ × {t} ∩ H_{t} with E =^{def} ℝ × ℝ·c̃. Then (exp W) ∩ H_{t,_{c}} for t ≠ k·π is just the

region of E below te graph of the function \{(f(x), x) | f(x) = −\frac{sin(2t)|_{c}|^{2}}{4(1−cos(2t))}·x^{2}\}. For t = k·π it is the half-ray ℝ^{1} × {0} (see Figure 2).

Figure 2: (exp W) ∩ H_{t,_{c}}.

We have choosen the parametrization of O_{4} in such a way that the oneparameter group exp(z, c, t) is entirely contained in the three-dimensional subspace ℝ × ℝ·c × ℝ. Since the image of W under the exponential function is invariant under the action of Ad(exp(0, 0, r)), which induces a rotation in {0} × ℂ × {0}, we can visualize exp W in ℝ × ℝ·c × ℝ as in Figure 3.

Figure 3: Intersection of exp W with ℝ × ℝ·c × ℝ.

3. The semigroup S = \overline{⟨exp W ⟩}

In this section we consider the closed semigroup S = \overline{⟨exp W ⟩} generated by W. We show that it is infinitesimally generated with tangent wedge L(S) = W. It turns out that exp W is not dense in S. Indeed, there are interior points in S \ exp W. This makes this example so important for the discussion of divisible semigroups and Lie semialgebras in [HR].

Lemma 3.1. Let B ⊆ O_{4} defined by B =^{def} ℝ × ℂ×] − π, π[. Then the restriction of the exponential map

exp |_{B}: B → B

is a diffeomorphism.

Proof. This follows from the explicit formula for the exponential map in Proposition 1.3.

Lemma 3.2. Let W be the invariant cone of Definition 2.2. Then the infinitesimally generated semigroup S = \overline{⟨exp W ⟩} satisfies (i) L(S) = W.

(ii) With B of Lemma 3.1 we have

S ∩ B = (exp W) ∩ B.

Proof. Let C = {((z, c, v), (z′, c′, v′)) ∈ B × B | −π ≤ r, r′, r + r′ ≤ π}. Then C = {(x, y) ∈ B × B | exp x· exp y ∈ B}. By Proposition 3.1, we have

exp sx· exp ty ∈ exp B für alle s, t ∈ [0, 1]

for all (x, y) ∈ C. Now [HHL] II.2.41 implies

exp x· exp y ∈ exp W for all (x, y) ∈ C ∩ (W × W).

Since [π, +∞[×ℂ × ℝ is a semigroup ideal in ℝ^{+} × ℂ × ℝ, this shows (ii). But then (i) is obvious.

With these preparations we can explicitly describe the semigroup S = \overline{⟨exp W ⟩}.

Proposition 3.3. Let O_{4} = ℝ×ℂ×ℝ be the oscillator group in the parametrization Lemma 1.2, and W the invariant Lorentz cone of Definition 2.2. Set S = S_{1} ∪ S_{2} with

S_{1} = \{(z, c, r) ∈ O_{4} | z ≤ −\frac{sin(2r)}{4(1 − cos(2r))}|c|^{2}\} ∪ ℝ^{−} × \{0\} × \{0\}

and

S_{2} = [π, +∞[×ℂ × ℝ.

Then

(i) S = \overline{⟨exp W ⟩},

(ii) L(S) = W.

Proof. (i) First, we show S ⊆ S_{∗} =^{def} \overline{⟨exp W ⟩}. By Proposition 3.2,

S_{1} = S_{∗} ∩ B = (exp W) ∩ B  (15)

holds. This implies S_{1} ⊆ S_{∗}. We have Φ_{c}(\frac{π}{2}) = exp \frac{π}{2}·(\frac{|c|^{2}}{4} , c, 1) = (0, c, \frac{π}{2}), and therefore \{0\} × ℂ × \{\frac{π}{2}\} ⊆ S_{∗}. For r ∈ ℝ^{+}, z ∈ ℝ^{−} the relations (0, 0, r) = exp(0, 0, r) ∈ S_{∗} and (z, 0, 0) = exp(z, 0, 0) ∈ S^{∗} hold. Because of

(π, v, 0)·(r, 0, 0) = (π + r, v, 0)  (16)

and

(r, v, 0)·(0, 0, z) = (r, v, z)  (17)

this implies ℝ^{−} × ℂ × [\frac{π}{2}, +∞[⊆ S_{∗}. By definition of S_{1} and Lemma 2.3, equation (16) inplies S \ (ℝ^{+} \ {0} × {0} × [π, +∞[) ⊆ S_{∗}.

It remains to show, that all points of the form (z, 0, π) with z ∈ ℝ^{+} \ {0} are in S_{∗}. Consider the one-parametergroups Φ_{c_{0}} and Φ_{e^{−2is}c_{0}} . Then we have

Φ_{c_{0}}(t)·Φ_{e^{−2is}c_{0}}(t) =

(−\frac{|c_{0}|^{2}}{4} · sin(2t) + \frac{1}{2} sin 2(t − s)·|c_{0}|^{2}, (e^{i(t−2s)} + e^{−it})·c_{0}, 2t).

In particular, for s − \frac{π}{2} = t = t_{0} we get

Φ_{c_{0}}(t_{0})·Φ_{e^{−2it_{0}}c_{0}}(t_{0}) = (−\frac{|c_{0}|^{2}}{4} · sin(2t_{0}), 0, 2t_{0}) ∈ S_{∗}.

Thus every element of the form (z, 0, π) with z ∈ ℝ^{+} \ {0} is approximable within S_{∗} by a suitable choose of t_{0} > \frac{π}{2} and c_{0} ∈ ℂ. Since S_{∗} is closed, this shows S ⊆ S_{∗}.

For the proof of (i) it remains to show that S is a semigroup. Obviously, SS_{2} ∪ S_{2}S ⊆ S holds. By (15) and Proposition 3.2, we also have S_{1}S_{1} ⊆ S. This proves the assertion. Claim (ii) now follows from Lemma 3.2.

Proposition 3.3 shows, that the entire half-space [π, +∞[×ℂ × ℝ is contained in S = \overline{⟨exp W ⟩}, whereas exp W misses open sets in [π, +∞[×ℂ × ℝ by Lemma 2.3.

References

[HHL] Hilgert, J., Hofmann, K. H., and Lawson, J. D., “Lie Groups, Convex Cones, and Semigroups”, Oxford Univ. Press, 1989.

[HR] Hofmann, K. H., and Ruppert, W. A. F., The structure of Lie groups which support closed divisible subsemigroups, Proc. Oberwolfach Meeting on Semigroups, July 1991.

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