Lobachevskii Journal of Mathematics Vol.4, 1999, 13–46

\begin{matrix}c \\ ◯ \end{matrix}H. Omori

H. Omori INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY

(submitted by B.N.Shapukov)

It is getting a major understanding that the function ring of the underlying space of the quantum world is not a commutative ring, and the underlying space itself is not even a point set.

In spite of this, the underlying space of the quantum world should be a continuum, on which one can make a calculus. We hope we can see such mathematics in the coming century.

This article is a survey of such effort or trial to make such calculus. However, this article does not give a bird-eye-view of the noncommutative world but this gives several toys which come from the noncommutative world.

§I Moyal product formula, Polar coordinate systems.

§II Noncommutative 2-sphere (formal version), The algebra of quantum groups.

§III μ-regulated algebras, Deformation quantizations.

§IV Berezin representation, Noncommutative 3-sphere.

§I.1 Moyal product formula

We start with a little general setting of noncommutative algebras. Consider a complete linear space A, and its dense linear subspace A′. A continuous linear mapping D : A → A is called a quasinilpotent mapping, if for every a ∈ A′ there exists n such that D^{n}a = 0.

A triple (A, A′, ∗_{0}) is called a two-sided A′-module, if the following two conditions are satisfied:

(1) (A′, ∗_{0}) is an algebra with respect to the multiplication ∗_{0} over the complex number C.

(2) For every a ∈ A′ the right multiplication ∗a and the left multiplication a∗ extend to continuous linear mappings of A into itself.

It is easy to see that if two of a, b, c are in A′, then the associativity holds, i.e. a ∗ (b ∗ c) = (a ∗ b) ∗ c.

A continuous linear mapping D : A → A is called a derivation if D(a∗_{0}b) = (Da) ∗_{0} b + a ∗_{0} (Db) holds for every a, b ∈ A′.

Let D, D′ be derivations of (A, A′, ∗_{0}) such that DD′ = D′D, and D, D′ are quasinilpotent mappings. Using D, D′ we define for an positive real number ℏ > 0 the product of Moyal type as follows (cf. [MO]):

f ∗_{1} g = fe^{ℏ\overleftarrow{D}∗_{0}\overrightarrow{D′}}g,  (1.1)

where e^{ℏ\overleftarrow{D}∗_{0}\overrightarrow{D′}} = ∑_{k≥0} \frac{(ℏ)^{k}}{k!} (\overleftarrow{D} ∗_{0} \overrightarrow{D′})^{k}, and leftarrow and rightarrow indicate to which element these operators act: i.e.

f(\overleftarrow{D} ∗_{0} \overrightarrow{D′})^{k}g = D^{k}f ∗_{0} D′^{k}g.

Then, it is easy to see that (A, A′, ∗_{1}) is a two-sided A′-module and D, D′ are derivations of the new module (A, A′, ∗_{1}). Apparently, the quasinilpotency is used only to ensure the convergence of the product formula. Hence, we can replace this condition to some weaker one, if these are considered under a suitable topology.

If D_{1} : (A, A′, ∗_{0}) → (A, A′, ∗_{0}) is a derivation such that [D_{1}, D] = [D_{1}, D′] = 0, then the mapping D_{1} : (A, A′, ∗_{1}) → (A, A′, ∗_{1}) is a derivation.

Lemma 1.1. Let (D_{1}, D′_{1}) is a pair of commuting derivations of (A, A′, ∗_{0}) such that D_{1}, D′_{1} are quasinilpotent, and D, D′, D_{1}, D′_{1} are mutually commutative. Define a new associative product ∗_{2} by

f ∗_{2} g = fe^{ℏ\overleftarrow{D_{1}}∗_{1}\overrightarrow{D′_{1}}}g.  (1.2)

Then, we have

f ∗_{2} g = fe^{ℏ(\overleftarrow{D}∗_{0}\overrightarrow{D′}+\overleftarrow{D_{1}}∗_{0}\overrightarrow{D′_{1}})}g  (1.3)

This may be written in the form

fe^{ℏ\overleftarrow{D_{1}}(e^{ℏ\overleftarrow{D}∗_{0}\overrightarrow{D′}})\overrightarrow{D′_{1}}}g = fe^{ℏ(\overleftarrow{D}∗_{0}\overrightarrow{D′}+\overleftarrow{D_{1}}∗_{0}\overrightarrow{D′_{1}})}g.  (1.4)

Setting D_{1} = −D, D′_{1} = D′ in (1.3), we have

f ∗_{0} g = fe^{−ℏ\overleftarrow{D}∗_{1}\overrightarrow{D′}}g.  (1.5)

This formula shows that one can reconstruct the original ∗_{0} by using ∗_{1}.

Setting D_{1} = −D′, D′_{1} = D and denoting

\overleftarrow{D} ∧^{∗_{0}} \overrightarrow{D′} = \overleftarrow{D} ∗_{0} \overrightarrow{D′} − \overleftarrow{D′} ∗_{0} \overrightarrow{D},

we see that the product

fˆ∗g = fe^{ℏ(\overleftarrow{D} ∧^{∗_{0}}\overrightarrow{D′})}g  (1.6)

defines a new two-sided A′-module. This product formula can be inverted to recapture the original ∗_{0}-product:

f ∗_{0} g = fe^{−ℏ(\overleftarrow{D}∧^{ˆ∗}\overrightarrow{D′})}g  (1.7)

Example, Weyl algebra:

Let U be an open subset of R^{2} and let (C^{∞}(U), C[x, y], ·) be the triple with ordinary polynomial ring and the ordinary commutative product. Define a ∗-product by

f ∗ g = fe^{(iℏ/2)(\overleftarrow{∂_{y}}∧^{·}\overrightarrow{∂_{x}})}g  (1.8)

This gives a new associative algebra (C[x, y], ∗), called the Weyl algebra, and makes (C^{∞}(U), C[x, y], ∗) a two-sided C[x, y]-module.

It is easy to see that

x ∗ y = x · y − \frac{iℏ}{2}, y ∗ x = x · y + \frac{iℏ}{2}.

Hence, we have [x, y] = −iℏ where [ , ] is the commutator: [a, b] = a∗b−b∗a. The Weyl algebra is nothing but the algebra generated by x, y with the fundamental relation [x, y] = −iℏ.

If we set z = x+iy, z̄ = x−iy, then we have the commutator [z, z̄] = −2ℏ. The algebra generated by z, z̄ with the fundamental relation [z, z̄] = −2ℏ is sometimes called the Wick algebra in physics. These are of course isomorphic.

∂_{x}, ∂_{y} act on (C^{∞}(U), C[x, y], ∗) as derivations, and these can be given as

∂_{x} = \frac{1}{iℏ}ad(y), ∂_{y} = −\frac{1}{iℏ}ad(x), where ad(a)(b) = [a, b].

Note that the original commutative product · can be recaptured by ∗-product:

f · g = fe^{−(iℏ/2)(\overleftarrow{∂_{y}}∧^{∗}\overrightarrow{∂_{x}})}g  (1.9)

We next remark that (C^{∞}(U), C[x, y], ∗) can be extended various ways:

(1) Let Σ^{0}(R^{2}) be the space of all C^{∞} functions on R^{2} with asymptotic expansion

f(x, y) ∼ f_{0}(θ) + f_{1}(θ)r^{−1} + f_{2}(θ)r^{−2} + ⋯, f_{i} ∈ C^{∞}(S^{1}),

where r^{2} = x^{2} + y^{2}. Then, ∗-product extends to (Σ^{0}(R^{2}), ∗), by using an integral expression of the product formula (cf. §IV.2 (4.2)).

(2) Let C^{∞}(R)[[y^{−1}]] be the space of all formal power series of y^{−1} with coefficients of C^{∞} functions on R:

f(x, y) = f_{0}(x) + f_{1}(x)y^{−1} + f_{2}(x)y^{−2} + · · · .

Then, ∗-product extends to (C^{∞}(R)[[y^{−1}]], ∗) by the same product formula (1.8).

(3) If ℏ is treated as a formal parameter, then ∗-product is defined on any open subset U of R^{2}. That is, by the same product formula, ∗-product is defined on C^{∞}(U)[[ℏ]]: the space of all formal power series of ℏ with coefficients in the space of all C^{∞} functions C^{∞}(U). This is because the Moyal product formula is bidifferential operator on each ℏ^{k}-component.

§I.2. Quantum polar coordinate systems

Let P(R_{+} × S^{1}; ρ, θ; ∘) be the space of all polynomials of ρ, ρ^{−1} and e_{∘}^{iθ}, where ∘ means the natural commutative product, and e_{∘}^{iθ} etc. means that the exponential function with respect to the ∘-product. We use the notation f_{∘}, if the function f is defined by using ∘-product.

The quantum polar coordinate system is the Weyl algebra P(R_{+} ×S^{1}; ρ, θ) such that product formula is given by

f ∗_{ℏ} g = fe^{− \frac{iℏ}{2} \overleftarrow{∂_{ρ}}∧^{∘}\overrightarrow{∂_{θ}}}g.  (1.10)

Denote this algebra by (P(R_{+} × S^{1}, ρ, θ), ∗_{ℏ}). By (1.10), we compute

\begin{aligned}e_{∘}^{iθ} ∗_{ℏ} ρ & = e_{∘}^{iθ}e^{\frac{iℏ}{2} \overleftarrow{∂_{θ}}∘\overrightarrow{∂_{ρ}}}ρ = (ρ − \frac{ℏ}{2}) ∘ e_{∘}^{iθ}, \\ ρ ∗_{ℏ} e_{∘}^{iθ} & = ρe^{− \frac{iℏ}{2} \overleftarrow{∂_{ρ}}∘\overrightarrow{∂_{θ}}}e_{∘}^{iθ} = (ρ + \frac{ℏ}{2}) ∘ e_{∘}^{iθ}. \end{aligned}  (1.11)

These give the exchange rule

ρ ∗_{ℏ} e_{∘}^{ikθ} = e_{∘}^{ikθ} ∗_{ℏ} (ρ + kℏ).  (1.12)

If we set z′ = \sqrt{2ρ} ∗_{ℏ} e^{iθ}, z̄′ = e^{−iθ} ∗_{ℏ} \sqrt{2ρ} by using a transcendental element \sqrt{2ρ}, then, we see z′ ∗_{ℏ} z̄′ = 2ρ, z̄′ ∗_{ℏ} z′ = 2ρ + 2ℏ. Hence [z′, z̄′] = −2ℏ. Note here that z′, z̄′ are invertible elements, because ρ and \sqrt{2ρ} can be treated invertible elements in the extended algebra.

Remark here that if [x, y] = −iℏ, then z = x + iy and z̄ = x − iy satisfies the same commutation relation [z, z̄] = −2ℏ, although z, z̄ are not invertible.

For every f ∈ C^{∞}(R) and for every integer k, the exchanging rule (1.12) gives

f_{∘}(ρ − kℏ) ∗_{ℏ} e_{∘}^{ikθ} = e_{∘}^{ikθ} ∗_{ℏ} f_{∘}(ρ)  (1.13)

Since ρ > 0, f_{∘}(ρ − kℏ) can not be defined if ρ − kℏ < 0 for f ∈ C^{∞}(R_{+}). Therefore, we see that (P(R_{+} × S^{1}; ρ, θ), ∗_{ℏ}) can not extend to the algebra involving C^{∞}(R_{+}).

By this reason, the variable ρ should be restricted in a neighborhood of ρ = ∞. This is for instance we have to treat ρ_{∘}^{−1} as a formal variable.

Now, once ρ_{∘}^{−1} is treated as a formal parameter, then (1.10) gives for every f_{∘}(θ) ∈ C^{∞}(S^{1}) that

\begin{matrix}f_{∘}(θ) ∗_{ℏ} ρ_{∘}^{−1} = ∑_{k≥0}(−\frac{iℏ}{2})^{k}∂^{k}f_{∘}(θ) ∘ ρ_{∘}^{−1−k} \\ ρ_{∘}^{−1} ∗_{ℏ} f_{∘}(θ) = ∑_{k≥0}(\frac{iℏ}{2})^{k}∂_{θ}^{k}f_{∘}(θ) ∘ ρ_{∘}^{−1−k}. \end{matrix}  (1.14)

Hence the ∗_{ℏ}-product is defined on the space C^{∞}(S^{1}) ∘ [ρ^{1/2}, ρ_{∘}^{−1/2}]] of all formal power series written in the form

f = ∑_{k=−∞}^{m} g_{k}(θ)_{∘} ∘ ρ_{∘}^{k/2},

for some integer m. The associativity is ensured automatically by polynomial approximations. That is, we have

Theorem 1.2. (C^{∞}(S^{1}) ∘ [ρ^{1/2}, ρ_{∘}^{−1/2}]], ∗_{ℏ}) is an associative algebra.

Here we remark the way of computation in this algebra. By (1.14), we have the exchanging rule

ρ_{∘}^{−1} ∗_{ℏ} e_{∘}^{iθ} = ∘\frac{ρ_{∘}^{−1}}{1 + \frac{ℏ}{2ρ}} e_{∘}^{iθ} = \{∘\frac{1 − \frac{ℏ}{2ρ}}{1 + \frac{ℏ}{2ρ}} ∘ e_{∘}^{iθ}\} ∗_{ℏ} ρ_{∘}^{−1}.

Hence we have for every f ∈ C^{∞}(S^{1}) the exchange rule

ρ_{∘}^{−1} ∗_{ℏ} f_{∗}(e_{∘}^{iθ}) = f_{∗} (∘\frac{1 − \frac{ℏ}{2ρ}}{1 + \frac{ℏ}{2ρ}} ∘ e_{∘}^{iθ}) ∗_{ℏ} ρ_{∘}^{−1}

where in the right hand side we take the Taylor expansion at the point e_{∘}^{iθ} to obtain formal power series of ρ_{∘}^{−1}.

By a similar computation for ρ_{∘}^{−1/2}, we get the similar exchanging rule. Thus, we can compute the ∗_{ℏ}-product.

Consider now the space Σ^{∞}(R^{2}; z, z̄): The space of all C^{∞} functions on R^{2} with asymptotic expansions

f ∼ ∑_{k=−∞}^{m} f_{k}(e_{·}^{iθ}) · r_{·}^{k}

where z = x + iy, and e_{·}^{iθ} = \frac{z}{\sqrt{z·z̄}}, r = \sqrt[·]{z · z̄}. Denote by Σ^{−s}(R^{2}; z, z̄) the subspace of Σ^{∞}(R^{2}; z, z̄) with asymptotic expansions starting at order r^{−s}.

As it mentioned in (1) in the previous section, we see the following:

Theorem 1.3. ∗_{ℏ}-product extends to the space Σ^{∞}(R^{2}; z, z̄) and makes this an associative algebra.

By the product formula (1.8), we see easily the product formula

f ∗_{ℏ} g = fe^{ℏ(\overleftarrow{∂z̄}∧^{·}\overrightarrow{∂_{z}})}g.

Using this product formula, we have

z̄ ∗_{ℏ} e_{·}^{− \frac{z·z̄+h(z̄)}{ℏ}} = 0 = e_{·}^{− \frac{z·z̄+h(z)}{ℏ}} ∗_{ℏ} z  (1.15)

for every holomorphic function h(z). Hence z, z̄ and e_{·}^{− \frac{z·z̄}{ℏ}} are not invertible in the ∗_{ℏ}-algebra, whenever ℏ is not a formal parameter. Recall z′ was invertible. Thus, we can not set z = z′ ( = \sqrt{2ρ} · e^{iθ}).

To simplify notations, we use in what follows the notation f ∗ g instead of f ∗_{ℏ} g.

To consider the correct relation between z and z′, we first compute the ∗-exponential e_{∗}^{− \frac{t}{2} z·z̄} = ∑ \frac{1}{k!}(−\frac{t}{2}z · z̄)_{∗}^{k} in Σ^{0}(R^{2}).

Lemma 1.4. For every ζ with Reζ ≥ 0 and for every ℏ ≥ 0, we have e_{∗}^{− \frac{ζ}{2} z·z̄} ∈ Σ^{−∞}(R^{2}), and

e_{∗}^{− \frac{ζ}{2} z·z̄} = \frac{2e^{ℏζ/2}}{e^{ℏζ} + 1} exp (−\frac{z · z̄}{ℏ}tanh\frac{ℏζ}{2} ) .

In particular, if t ∈ R, then lim_{t→∞} e_{∗}^{− \frac{t}{2} z̄∗z} = 0, but

lim_{t→∞} e_{∗}^{− \frac{t}{2} z∗z̄} = lim_{t→∞} \frac{2e^{tℏ}}{e^{tℏ} + 1} exp (−\frac{z · z̄}{ℏ}tanh\frac{tℏ}{2}) = 2e^{− \frac{z·z̄}{ℏ}}

Proof. Set f_{t} = e_{∗}^{−t \frac{1}{2} zz̄}. Suppose f_{t} = F_{t}(z · z̄). By the product formula (1.8), we have

\begin{matrix}\frac{∂}{∂t}F_{t}(z · z̄) = −\frac{1}{2}(zz̄) ∗ F_{t}(z · z̄) \\ = −\frac{1}{2}(z · z̄) · F_{t}(z · z̄) + \frac{ℏ^{2}}{2} F′_{t}(z · z̄) + \frac{ℏ^{2}}{2} z · z̄ · F′′_{t}(z · z̄). \end{matrix}

F_{t} must satisfy the differential equation

\frac{∂}{∂t}F_{t}(s) = −\frac{1}{2}sF_{t}(s) + \frac{ℏ^{2}}{2} (F′_{t}(s) + sF′′_{t}(s))

with the initial condition F_{0}(s) = 1.

To find the solution, we set first F_{t}(s) = g(t) exp\{−\frac{s}{ℏ}tanh\frac{ℏt}{2}\}. Then, g(t) must satisfy the equation

g′(t) = −\frac{ℏ}{2} (tanh\frac{ℏt}{2}) g(t), g(0) = 1.

Hence, we have g(t) = \frac{1}{cosh (ℏt/2)}. □

Using Lemma 1.4, we have the following:

Lemma 1.5. Since ℏ is positive, λ+z̄∗z = λ+ℏ+zz̄ is invertible for every λ > −ℏ and

(λ + z̄ ∗ z)^{−1} = ∫_{0}^{∞}e^{−λt}e_{∗}^{−tz̄∗z}dt.

Proof. Clearly, ∫_{0}^{∞}e^{−λt}e_{∗}^{−tz̄∗z}dt exists. We compute

\begin{matrix}(λ + z̄ ∗ z) ∗ ∫_{0}^{∞}e^{−λt}e_{∗}^{−tz̄∗z}dt = −∫_{0}^{∞} \frac{d}{dt}(e^{−λt}e_{∗}^{−tz̄∗z})dt \\ = 1 − lim_{t→∞} e^{−λt}e_{∗}^{−tz̄∗z} = 1. \end{matrix}

We can see that z̄ ∗ z is invertible but z ∗ z̄ is not invertible.

Using the formula of Laplace transform, we define ;

\sqrt[∗]{\frac{π}{z̄∗z}} = ∫_{0}^{∞}\sqrt{t}^{−1}e_{∗}^{−tz̄∗z}dt.  (1.16)

Noting that

H(zz̄) ∗ z = (H(zz̄) − ℏH′(zz̄))z,

we can compute z ∗ \sqrt[∗]{z̄ ∗ z}^{−1}.

Lemma 1.6. z ∗ \sqrt[∗]{z̄ ∗ z}^{−1} ∈ Σ^{0}(R^{2}) such that

z ∗ \sqrt[∗]{z̄ ∗ z}^{−1} = \sqrt[.]{z · z̄}^{−1} · z · (1 + ℏh(z · z̄))

Proof. Use the identity z ∗ \sqrt[∗]{z̄ ∗ z}^{−1} = π^{− \frac{1}{2}} ∫_{0}^{∞}\sqrt{t}^{−1}z ∗ e_{∗}^{−tz̄∗z}dt.

For the computation, the following lemma is very powerful:

Bumping Lemma. For a function f(t) defined by a power series of t, we define f_{∗}(z ∗ z̄), f_{∗}(z̄∗ z) by the same power series. If these are well defined, then the equalities

f_{∗}(z ∗ z̄) ∗ z = z ∗ f_{∗}(z̄ ∗ z), z̄ ∗ f_{∗}(z ∗ z̄) = f_{∗}(z̄ ∗ z) ∗ z̄

hold in Σ^{0}(R^{2}). Hence we have

f_{∗}(zz̄ − ℏ) ∗ z = z ∗ f_{∗}(zz̄ + ℏ).

Proof. For the proof we remark at first that the equality (z ∗ z̄)^{n} ∗ z = z ∗ (z̄ ∗ z)^{n}, and hence e_{∗}^{tz∗z̄} ∗ z = z ∗ e^{tz̄∗z} etc.. □

For such functions f and g of one variable, we have

f_{∗}(z ∗ z̄) ∗ g_{∗}(z ∗ z̄) = (fg)_{∗}(z ∗ z̄).  (1.17)

Using (1.16), we define ρ̂∗^{\frac{1}{2}} = \sqrt[∗]{z̄ ∗ z} by (z̄ ∗ z) ∗ \sqrt[∗]{z̄ ∗ z}^{−1}, and set τ̂ = e_{∗}^{− \frac{1}{2} z∗z̄}, T = z ∗ \sqrt[∗]{z̄ ∗ z}^{−1}, T^{∗} = \sqrt[∗]{z̄ ∗ z}^{−1} ∗ z̄. Then, denoting

ϖ = 2e^{− \frac{z·z̄}{ℏ}} = lim_{t→∞} e_{∗}^{− \frac{t}{2} z∗z̄},

we have Lemma 1.7. ρ̂ = z̄ ∗ z, τ̂, T, T^{∗} and ϖ satisfy the following:

[ρ̂, T] = 2ℏT, τ̂^{m} ∗ T = e^{−mℏ}T ∗ τ̂^{m}.
T^{∗} ∗ T = 1, T ∗ T^{∗} = 1 − ϖ, ϖ ∈ Σ^{−∞}(R^{2}, z, z̄),

and ϖ ∗ ϖ = ϖ, T^{∗} ∗ ϖ = ϖ ∗ T = 0.

In fact, T, T^{∗} generate the algebra called Töplitz algebra.

Let I be the two sided ideal of Σ^{∞}(R^{2}) generated by ϖ. Then we have

Theorem 1.8. By the correspondence ρ̂^{\frac{1}{2}} ↦ ρ^{\frac{1}{2}} , T ↦ e^{iθ}, we have

(Σ^{∞}(R^{2})/I, ∗) ≅ (C^{∞}(S^{1}) ∘ [ρ^{1/2}, ρ_{∘}^{−1/2}]], ∗).

This gives the correct correspondence between two algebras. Note that the algebra (C^{∞}(S^{1})∘[ρ^{1/2}, ρ_{∘}^{−1/2}]], ∗) can be viewed as a noncommutatization or quantization of the commutative algebra C^{∞}(S^{1}) ∘ [ρ^{1/2}, ρ_{∘}^{−1/2}]]. Thus, the algebra (Σ^{∞}(R^{2}), ∗) is twice noncommutatized from C^{∞}(S^{1}) ∘ [ρ^{1/2}, ρ_{∘}^{−1/2}]].

Remark. So far we have treated ℏ as a positive parameter. If ℏ is treated as a formal parameter, then as it is mentioned in §I.1, the ∗-products are well defined on spaces

C^{∞}(R^{2})[[ℏ]], and C^{∞}(R_{+} × S^{1})[[ℏ]]

and these algebra can be localized on every open subsets by the same product formulas (cf. Theorem 3.1).

If we localize C^{∞}(R^{2})[[ℏ]] to the open subset R^{2} − {0}, then z, z̄ are invertible inside (C^{∞}(R^{2} − {0})[[ℏ]], ∗), and hence ρ = z̄∗ z, and the unitary part of z, i.e. e^{iθ} = z ∗ \sqrt[∗]{z̄ ∗ z}^{−1} are well-defined. Hence we see (C^{∞}(R^{2} − {0})[[ℏ]], ∗) ≅ (C^{∞}(R_{+} × S^{1})[[ℏ]], ∗)

§II.1 Noncommutative 2-sphere (formal version)

Let 𝔤 = 𝔬(3) be the Lie algebra generated by three elements x_{1}, x_{2}, x_{3} with the following commutation relations:

[x_{1}, x_{2}] = x_{3}, [x_{2}, x_{3}] = x_{1}, [x_{3}, x_{1}] = x_{2}.

We regard x_{i} linear functions on the dual space 𝔤^{∗} ≅ R^{3} of 𝔤. Let C^{∞}(𝔤^{∗}) be the space of all C^{∞} functions on 𝔤^{∗}. For every f, g ∈ C^{∞}(𝔤^{∗}), we define a bilinear operation { , }, called the Poisson bracket, by

{f, g}(p) = [(df)_{p}, (dg)_{p}](p), p ∈ 𝔤^{∗}.

This gives the structure of Poisson algebra on the space C^{∞}(𝔤^{∗}). That is,

(a) {f, g} = −{g, f}, (b) {fg, h} = f{g, h} + {f, h}g,

(c) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0.

It is known that there exists an associative algebra (C^{∞}(𝔤^{∗})[[ν]], ∗) such that

(1) ν ∈ center and treated as a formal parameter. (Cf. the remark at the end of this section for the case that ν is not a formal parameter.)

(2) for every f, g ∈ C^{∞}(𝔤^{∗})

f ∗ g = fg + \frac{ν}{2} \{f, g\} + ν^{2}π_{2}(f, g) + ⋯,  (2.1)

(3) π_{k} are bidifferential operators,

(4) π_{k}(x_{i}, x_{j}) = 0 for k ≥ 2. (Cf. [OMY2].)

From the properties (1), (2) we see that the noncommutative algebra (C^{∞}(𝔤^{∗})[[ν]], ∗) involves the Poisson algebra (C^{∞}(𝔤^{∗}), ·, { , }) in the first and second components of (2.1).

By the property (3), this algebra can be localized on every open subset of 𝔤^{∗}. We see

[x_{i}, x_{j}]_{∗} = νx_{k}, (cyclic) i, j, k, x_{i} ∗ x_{i} = x_{i}x_{i}, i = 1, 2, 3.

(C^{∞}(𝔤^{∗})[[ν]], ∗) involves the universal enveloping algebra of the Lie algebra 𝔤.

We have r^{2} = x_{1} ∗ x_{1} + x_{2} ∗ x_{2} + x_{3} ∗ x_{3} = x_{1}x_{1} + x_{2}x_{2} + x_{3}x_{3} ∈ center.

Localize this algebra where r^{−1} exists. Precisely speaking, we consider the algebra C^{∞}(𝔤^{∗} − {0})[[ν]]. Set μ = νr^{−1} and let (C^{∞}(S^{2})[[μ]], ∗) be the subalgebra generated topologically by μ and

(X_{1}, X_{2}, X_{3}) = (x_{1} ∗ r^{−1}, x_{2} ∗ r^{−1}, x_{3} ∗ r^{−1})

where “topologically” means we take the topological closure after making the algebra generated by algebraic procedures. We have easily that [X_{i}, X_{j}] = μX_{k} for cyclic i, j, k and

X_{1} ∗ X_{1} + X_{2} ∗ X_{2} + X_{3} ∗ X_{3} = X_{1}X_{1} + X_{2}X_{2} + X_{3}X_{3} = 1

Set ζ = X_{1} + iX_{2}, ρ = X_{3}.

Then, we have

ζ ∗ ζ̄ = 1 − ρ^{2} − iμρ, ζ̄ ∗ ζ = 1 − ρ^{2} + iμρ,
[ρ, ζ] = −iμζ, [ρ, ζ̄] = iμζ̄, [ρ, ζ ∗ ζ̄] = 0.

Localize this algebra to the open set \{ρ ≠ ±1\}. Then, elements \sqrt{ζ ∗ ζ̄}^{−1}, \sqrt{ζ̄ ∗ ζ}^{−1} exist in the localized algebra.

Set e^{iθ} = \sqrt[∗]{ζ ∗ ζ̄}^{−1} ∗ζ. This is the unitary part of ζ, since μ is treated as a formal parameter. We can see easily that [ρ, e^{iθ}] = iμe^{iθ}. This is equivalent to say that [ρ, θ] = μ.

Since ρ = ±1 corresponds to north and south poles, the algebra localized to C^{∞}(S^{2} − {N, S})[[μ]] is written by ρ, θ, and the algebra (C^{∞}(S^{2} − {N, S})[[μ]], ∗) is isomorphic to C^{∞}(S^{1} × (−1, 1))[[μ]] with ∗-product

f ∗ g = fe^{^{\frac{μ}{2}} \overleftarrow{∂_{ρ}}∧\overrightarrow{∂_{θ}}}g,

where μ is treated as formal parameter.

By the stereographic projection from the north pole, we set

z = \sqrt{\frac{1 + ρ}{1 − ρ}} ∗ e^{iθ}.

Then, since [ρ, e^{iθ}] = iμe^{iθ} we have

[z, z̄] = μ(1 + z ∗ z̄) ∗ (1 + z̄ ∗ z)  (2.2)

By the stereographic projection from the south pole, we set

w = e^{−iθ} ∗ \sqrt{\frac{1 − ρ}{1 + ρ}}.

Then,

[w, w̄] = μ(1 + w ∗ w̄) ∗ (1 + w̄ ∗ w)  (2.3)

Since two variables are related by w = z_{∗}^{−1}, C^{∞}(S^{2})[[μ]] can be viewed as a patchwork of “local” coordinate algebras,

(C^{∞}(C)[[μ]], ∗; z, z̄), and (C^{∞}(C)[[μ]], ∗; w, w̄).

Remark that the coordinate transformation w = z_{∗}^{−1} can be viewed as a holomorphic diffeomorphism, in the sense that this does not involve the variables z̄, w̄.

Remark. So far, we have treated μ as a formal parameter. In the last part of this section, we give a remark for the case that μ is not a formal parameter. Let Σ^{0}(R^{3}) ⊂ C^{∞}(𝔤^{∗}) be the subspace of all functions with asymptotic expansion

f ∼ f_{0}(θ) + f_{1}(θ)r^{−1} + f_{2}(θ)r^{−2} + ⋯, where f_{i} ∈ C^{∞}(S^{2}).

Let Σ^{−k}(R^{3}) is the subspace such that f_{0} = f_{1} = ⋯ f_{k−1} = 0. It is easy to see that

Σ^{−k}(R^{3})Σ^{−l}(R^{3}) ⊂ Σ^{−k−l}(R^{3}).

Recall μ = \frac{ν}{r}. Identifying μ with r^{−1} gives

C^{∞}(S^{2})[[μ]] ≅ Σ^{0}(R^{3})/Σ^{−∞}(R^{3}).

Thus, as in the case R^{2}, it is natural to ask whether there exists a ∗- product on the space Σ^{0}(R^{3}) so that

Σ^{−k}(R^{3}) ∗ Σ^{−l}(R^{3}) ⊂ Σ^{−k−l}(R^{3})

and (Σ^{0}(R^{3})/Σ^{−∞}(R^{3}), ∗) ≅ (C^{∞}(S^{2})[[μ]], ∗). In fact, we have the following:

Theorem 2.1. Suppose iν is a positive real parameter. Then, there is no such a ∗-product.

Proof. Now suppose there exists such a ∗-product on Σ^{0}(R^{3}). If we set

X = \frac{x_{1}}{\sqrt{1 + r^{2}}} , Y = \frac{x_{2}}{\sqrt{1 + r^{2}}} , Z = \frac{x_{3}}{\sqrt{1 + r^{2}}} .

Then X, Y, Z ∈ Σ^{0}(R^{3}) and

X^{2} + Y^{2} + Z^{2} = \frac{r^{2}}{1 + r^{2}} , [X, Y] = \frac{ν}{\sqrt{1 + r^{2}}} Z ( cyclic).  (2.4)

We see easily that f(X, Y, Z) ∈ Σ^{0}(R^{3}) for every C^{∞} function on R^{3}. Using the commutation relation, we have

\begin{aligned}(X + iY) ∗ Z & = (Z + \frac{iν}{\sqrt{1 + r^{2}}} ) ∗ (X + iY), \\ (X + iY) ∗ Z^{k} & = (Z + \frac{iν}{\sqrt{1 + r^{2}}} )^{k} ∗ (X + iY). \end{aligned}  (2.5)

Set ϕ(Z) = 1/(1 − _{\frac{Z}{_{\frac{r}{\sqrt{1+r^{2}}}} +iν}}). Since iν is a positive real parameter, ϕ(Z) ∈

Σ^{0}(R^{3}) and ϕ(Z) is the limit function of ϕ_{k}(Z) = ∑_{i=0}^{k}(_{\frac{Z}{_{\frac{r}{\sqrt{1+r^{2}}}} +iν}})^{i}. By

(2.4) and (2.5), we have

(X + iY) ∗ ϕ_{k}(Z) ∗ (X − iY) = ϕ_{k}(Z + \frac{iν}{\sqrt{1 + r^{2}}} ) ∗ (\frac{r^{2}}{1 + r^{2}} − Z^{2} − \frac{2iν}{\sqrt{1 + r^{2}}} Z).

(2.6)

Since lim_{k→∞} ϕ_{k}(Z) = ϕ(Z), we see that (X + iY) ∗ ϕ(Z) ∗ (X − iY) makes sense in the left hand side. However, The right hand side of (2.6) tends to ∞ at Z = \frac{r}{\sqrt{1+r^{2}}} . This is a contradiction. □

§II.2 Algebra of quantum group SU_{q}(2)

In this section, we give another interesting example. Consider the algebra A(SU_{q}(2)) generated by the following four elements X, U, V, Y, with the relations:

U ∗ X = qX ∗ U, V ∗ X = qX ∗ V,
Y ∗ U = qU ∗ Y, Y ∗ V = qV ∗ Y,
U ∗ V = V ∗ U
X ∗ Y − q^{−1}U ∗ V = Y ∗ X − qU ∗ V = 1.

This is called the coordinate ring of the quantum group SU_{q}(2).

Though A(SU_{q}(2)) has also a co-algebra structure, we only concern with the algebra of A(SU_{q}(2)). To understand this algebra, we first consider coordinate functions X, Y, U, V of the group SU(2): Set Y = X̄, V = −Ū

( \begin{matrix}X & U \\ −Ū & X̄ \end{matrix} ) , with XX̄ + UŪ = 1

SU(2) ≈ S^{3}, and this is covered by two coordinate sheets as follows: One is C × S^{1} with the correspondence

( \begin{matrix}X & U \\ V & Y \end{matrix} ) = ( \begin{matrix}\sqrt{\frac{1−e_{.}^{−z·z̄}}{zz̄}} · z & ie^{− \frac{1}{2} z·z̄} · e^{iϑ} \\ ie_{.}^{− \frac{1}{2} z̄·z}e_{.}^{−iϑ} & \sqrt{\frac{1−e^{−z·z̄}}{zz̄}} · z̄ \end{matrix}) .

This sheet covers an open dense subset of SU(2). The set which is not covered is given by z = ∞: the set D = diag{e^{iθ}, e^{−iθ}}.

Near the subset D, we use another coordinate sheet:

( \begin{matrix}X & U \\ V & Y \end{matrix} ) = (\begin{matrix}\sqrt[·]{1 − τ^{2}} · e^{iθ} & iτe^{iϑ} \\ iτ · e^{−iϑ} & \sqrt[·]{1 − τ^{2}} · e^{−iθ} \end{matrix} ) , 0 ≤ τ < 1,

where ζ = τ · e^{iϑ} is viewed as the coordinate function of C near the origin. Hence the coordinate transformation is

\begin{matrix}τ = e_{.}^{− \frac{1}{2} zz̄}, \\ e^{iθ} = \sqrt[.]{z · z̄}^{−1} · z, \\ e^{iϑ} = e_{.}^{iϑ} \end{matrix}  (2.7)

Since the coordinate z, z̄ covers the domain 0 < τ < 1, one may restrict the domain of τ, e^{iθ} coordinate to an infinitesimal neighborhood of τ = 0. This means we treat the function ring of all formal power series in ρ with coefficients of C^{∞} functions of e^{iθ}, e^{iϑ}.

Let Σ^{0}(C × S^{1}) be the space of all C^{∞} function f with asymptotic expansion

f ∼ f_{0}(e^{iθ}, e^{iϑ}) + f_{1}(e^{iθ}, e^{iϑ})r^{−1} + f_{2}(e^{iθ}, e^{iϑ})r^{−2} + ⋯

where z = re^{iθ}. Define ∗-product by the Moyal product formula

f ∗ g = fe^{ℏ\overleftarrow{∂_{z}}∧\overrightarrow{∂z̄}}g

where e^{iϑ} is in the center. Remark that [z, z̄] = 2ℏ under this product. (In §II.2, we used the product such that [z, z̄] = −2ℏ.) We see that (Σ^{0}(C × S^{1}), ∗) is an associative algebra and here ℏ is not treated as a formal parameter.

In this algebra, we set as follows:

\begin{matrix}X = \sqrt[∗]{\frac{1 − e_{∗}^{−z∗z̄}}{z ∗ z̄}} ∗ z, Y = z̄ ∗ \sqrt[∗]{\frac{1 − e_{∗}^{−z∗z̄}}{z ∗ z̄}} , \\ U = ie_{∗}^{− \frac{1}{2} z∗z̄} ∗ e^{iϑ}, V = ie_{∗}^{− \frac{1}{2} z̄∗z} ∗ e^{−iϑ} \end{matrix}  (2.8)

the inside of √ is computed as a function of one variable t = z ∗ z̄ and hence this expression makes sense at t = 0. Remark here the delicate difference that z ∗ z̄ and z̄ ∗ z are used in the definition of U and V. Remark also that

U = ie^{− \frac{ℏ}{2}} e_{∗}^{− \frac{1}{2} zz̄}e^{iϑ}, V = ie^{\frac{ℏ}{2}} e_{∗}^{− \frac{1}{2} zz̄}e^{−iϑ}

and hence U ∗ V = −e_{∗}^{−zz̄}.

Using these formulas and setting q = e^{ℏ}, we see that the elements X, Y, U, V satisfy the relations of generators of A(SU_{q}(2)).

For instance, using bumping lemma, we have

Y ∗ X = z̄ ∗ \frac{1 − e_{∗}^{−z∗z̄}}{z ∗ z̄} ∗ z = z̄ ∗ z ∗ \frac{1 − e_{∗}^{−z̄∗z}}{z̄ ∗ z} = 1 − e_{∗}^{−zz̄}e^{ℏ} = 1 + qU ∗ V.

V ∗ X =ie_{∗}^{− \frac{1}{2} z̄∗z} ∗ e^{−iϑ} ∗ \sqrt{\frac{1 − e_{∗}^{−z∗z̄}}{z ∗ z̄}} ∗ z

=\sqrt{\frac{1 − e_{∗}^{−z∗z̄}}{z ∗ z̄}} ∗ ie_{∗}^{− \frac{1}{2} z∗z̄+ℏ} ∗ e^{−iϑ} ∗ z = qX ∗ V.

Thus, we see that there is a homomorphism of the algebra A(SU_{q}(2)) into

(Σ^{0}(R^{2} × S^{1}), ∗).

On the other hand, let P(S^{1} × S^{1}) be the polynomial ring of e^{±iθ}, e^{±iϑ}. On P(S^{1} × S^{1})[[τ]] define a ∗-product by the product formula

f ∗ g = fe^{^{\frac{iℏ}{2}} \overleftarrow{τ∂_{τ}}∧\overrightarrow{∂_{θ}}}g.  (2.9)

If we set τ = e^{ρ}, then τ∂_{τ} = ∂_{ρ}. Hence, (2.9) is essentially the Weyl algebra. Let e^{iϑ} be the variable moving in S^{1}. e^{iϑ} commutes with other elements.

In (P(S^{1} × S^{1})[[τ]], ∗), we set as follows:

X = \sqrt[·]{1 − τ^{2}} · e^{iθ}, Y = \sqrt[·]{1 − τ^{2}} · e^{−iθ}
U = iτ · e^{\frac{−ℏ}{2}} · e^{iϑ}, V = iτ · e^{\frac{ℏ}{2}} · e^{−iϑ}.  (2.10)

where \sqrt[·]{1 − τ^{2}} stands for its Taylor expansion at τ = 0.

By these relations we have the following:

Lemma 2.2. X, Y, U, V satisfy the fundamental relation of the coordinate ring of A(SU_{q}(2)), i.e.

\begin{matrix}X ∗ Y = 1 + q^{−1}U ∗ V, Y ∗ X = 1 + qU ∗ V \\ X ∗ U = q^{−1}U ∗ X, X ∗ V = q^{−1}V ∗ X. \end{matrix}

Similarly we have

Y ∗ U = qU ∗ Y, Y ∗ V = qV ∗ Y, U ∗ V = V ∗ U.

where q = e^{ℏ}.

Proof.

X = \sqrt[·]{1 − e^{−ℏ}τ^{2}} ∗ e^{iθ}, Y = e^{−iθ} ∗ \sqrt[·]{1 − e^{−ℏ}τ^{2}}.

Hence we have X ∗ Y = 1 − e^{−ℏ}τ^{2} = 1 + q^{−1}U ∗ V.

Similarly, we have Y ∗ X = 1 − e^{ℏ}τ^{2} = 1 + qU ∗ V. For the third equality, we compute as follows:

\begin{matrix}X ∗ U = ie^{\frac{−ℏ}{2}} e^{iϑ} · \sqrt[·]{1 − τ^{2}} · e_{·}^{iθ} ∗ τ = ie^{\frac{−ℏ}{2}} e^{iϑ} · τ ∗ \sqrt[·]{1 − τ^{2}}e^{iθ−ℏ} \\ = q^{−1}ie^{\frac{−ℏ}{2}} e^{−iϑ} · τ ∗ \sqrt[·]{1 − τ^{2}}e^{iθ} = q^{−1}U ∗ X. \end{matrix}

□

By comparing (2.8) with (2.10), we see that (z, z̄, ϑ) and (θ, τ, ϑ) are related as follows:

τ = e_{∗}^{− \frac{1}{2} zz̄} = \frac{2e^{\frac{ℏ}{2}}}{e^{ℏ} + 1} exp (−\frac{zz̄}{ℏ}tanh\frac{ℏ}{2})

e^{iθ} = \sqrt[∗]{z ∗ z̄}^{−1} ∗ z mod ϖ, (Cf.§I.2)

ϑ = ϑ,

(2.11)

where ϖ = lim_{t→∞} e_{∗}^{− \frac{t}{2} zz̄} = lim_{k→∞} τ_{∗}^{k}. If ℏ = 0, then we have (2.7).

On the other hand, consider the topological closure of P(S^{1} × S^{1})[[τ]]. To do that we remark first that

τ^{m} ∗ e^{iθ} = e^{− \frac{mℏ}{2} e^{iθ}}τ^{m}, e^{iθ} ∗ τ^{m} = e^{\frac{mℏ}{2} e^{iθ}}τ^{m}.  (2.12)

Hence, for every p ∈ P(S^{1} × S^{1})[[τ]] we have the exchanging rule

τ^{m} ∗ p(e^{iθ}) = p(e^{− \frac{mℏ}{2} e^{iθ}})τ^{m}, p(e^{iθ}) ∗ τ^{m} = p(e^{\frac{mℏ}{2} e^{iθ}})τ^{m}.  (2.13)

Let C(S^{1} × S^{1}) be the space of all f ∈ C^{∞}(S^{1} × S^{1}) such that f(e^{mℏ}e^{iθ}, e^{iϑ}) is defined for all m ∈ Z. By a similar argument as in the proof of §I, Theorem 1.2, we see that the algebra (C(S^{1} × S^{1})[[τ]], ∗) is defined.

Now, we define an algebra Ã(SU_{q}(2)) which contains A(SU_{q}(2)) as a dense subalgebra as a patchwork of “local” coordinate algebras, where by local coordinate algebras we mean algebras that were mentioned above:

(Σ^{0}(C × S^{1}), ∗) and (C(S^{1} × S^{1})[[τ]], ∗)

We define f̃ ∈ Ã(SU_{q}(2)) as a pair f̃ = (f, g) such that f ∈ Σ^{0}(C×S^{1}), g ∈ C(S^{1} ×S^{1})[[τ]], and f, g are mutually translated through coordinate transformation (2.11). By the componentwise product, Ã(SU_{q}(2)) forms an algebra.

Remark that every g ∈ C(S^{1} × S^{1})[[τ]] can be written in the form

g ∼ g_{0} + g_{1} ∗ τ + g_{2} ∗ τ^{2} + ⋯, g_{k} ∈ C(S^{1} × S^{1}).

According to (2.11), we replace inside of f, τ by τ̂ = e_{∗}^{− \frac{1}{2} zz̄} and e^{iθ} by T = \sqrt[∗]{z ∗ z̄}^{−1} ∗ z. Then, obtained element g(T, e^{iϑ}, τ̂) must be an element of Σ^{0}(C × S^{1}), which has an asymptotic expansion

g(T, e^{iϑ}, τ̂) ∼ g_{0}(T, e^{iϑ}) + g_{1}(T, e^{iϑ}) ∗ τ̂ + g_{2}(T, e^{iϑ}) ∗ τ̂^{2} + ⋯,

Since ϖ = ϖ ∗ τ̂^{m} = lim_{m} τ̂^{m}, the exchange rule (2.12) yields

\begin{matrix}g(T, e^{iϑ}, τ̂) ∗ ϖ = lim_{m} g(T, e^{iϑ}, τ̂) ∗ τ̂^{m} = ∑ g_{k}(∞, e^{iϑ})ϖ \\ ϖ ∗ g(T, e^{iϑ}, τ̂) = lim_{m} τ̂^{m} ∗ g(T, e^{iϑ}, τ̂) = ∑ g_{k}(0, e^{iϑ})ϖ. \end{matrix}  (2.14)

Hence g must have the property that (2.13) makes sense. This shows that the space Ã(SU_{q}(2)) is much restricted.

§III.1 μ-regulated algebras

A complete topological associative algebra (A, ∗) with 1 and with an involutive anti-automorphism a ↦ ā is called a μ-regulated algebra (cf.[OMMY]), if (A.1) There exists μ ∈ A, called a regulator, such that μ̄ = μ, and [μ, A] ⊂

μ∗A∗μ, where [a, b] = a∗b − b∗a, the commutator bracket. (A.2) [A, A] ⊂ μ∗A. (A.3) There exists a closed linear subspace B such that A = B ⊕ μ ∗ A

(topological direct sum). (A.4) The mappings μ∗ : A → μ∗A, ∗μ : A → A∗μ defined by a → μ ∗ a,

a → a ∗ μ respectively are linear isomorphisms. (Self-similarity.) The following is easy to see:

By (A.1-2): μ ∗ A = A ∗ μ and 2-sided ideal. ∗ defines a commutative associative product · on A/μ ∗ A ≅ B.

In (A.3): B can be assumed \overline{B} = B.

By (A.3): for every N > 0, A is decomposed into:

A = B ⊕ μ∗B ⊕ ⋯ ⊕ μ^{N−1}∗B ⊕ μ^{N} ∗A.

(A, ∗) is called to be formal if ⋂_{k} μ^{k}∗A = {0}.

Every formal μ-regulated algebra is continuously embedded in the direct product

∏_{k=0}^{∞}μ^{k}∗B i.e. A ⊂ B[[μ]].

In what follows we concerns mainly the case A = B[[μ]].

Remark that one can change the filtration A = B[[μ]] by a linear isomorphism

L̃ : B[[μ]] → B[[μ]].

For instance for any continuous linear mapping L : B → B, we define a linear isomorphism L̃ : B[[μ]] → B[[μ]] by

L̃(f)=f + μ ∗ L(f), f ∈ B, L̃(μ)=μ.

L̃^{−1} is obtained by successive substitution to the equality f = g − μ ∗ L(f).

By (A.4), μ^{−1} defines a derivation of A on any μ-regulated algebra (A, ∗), i.e. [μ^{−1}, a] is defined by [μ^{−1}, a] = −μ^{−1} ∗ [μ, a] ∗ μ^{−1}. We see [μ^{−1}, A] ⊂ A, i.e. ad(μ^{−1}) is a derivation of (A, ∗), and hence ad(μ^{−1}) induces a derivation of (B, ·).

The product ∗ on B[[μ]] extends by (A.4) to the space A[μ^{−1}] : the space consisting of elements

μ^{−k} ∗ a_{−k} + ⋯ + μ^{−1} ∗ a_{−1} + a_{0} + μ ∗ a_{1} + μ^{2} ∗ a_{2} + · · · .  (3.1)

By these relations (A[μ^{−1}], ∗) is an associative algebra involving A as a subalgebra.

For every a ∈ A, we see that [μ^{−1}∗a, A] ⊂ A and hence ad(μ^{−1}∗a) gives a derivation of (A, ∗). We see also that

[μ^{−1} ∗ A, μ^{−1} ∗ A] ⊂ μ^{−1} ∗ A  (3.2)

i.e. μ^{−1} ∗ A forms a Lie algebra.

Note that every element of an FμR-algebra can be written uniquely in the form

a = a_{0} + μ ∗ a_{1} + μ^{2} ∗ a_{2} + ⋯, a_{i} ∈ B.  (3.3)

Thus, for every a, b ∈ B, the product a ∗ b and ad(μ^{−1})a are decomposed into the following form:

\begin{matrix}a ∗ b = a · b + μ ∗ π_{1}(a, b) + ⋯ + μ^{k} ∗ π_{k}(a, b) + ⋯ \\ ad(μ^{−1})a = ξ_{0}(a) + ⋯ + μ^{k} ∗ ξ_{k}(a) + · · · . \end{matrix}

Example 1 (Cf. §I.1 and §I.2)

(Σ^{0}(R^{2}), ∗) is a μ-regulated algebra. where μ^{−1} = ρ = \frac{zz̄}{2}. (C^{∞}(S^{1})[[μ]], ∗) is a formal μ-regulated algebra.

Example 2 (Cf. §II.1)

(C^{∞}(S^{2})[[μ]], ∗) is a formal μ-regulated algebra. This is also an example of quantized symplectic algebra, where the definition will be given Definition 3.2.

Example 3

Here we give another example which we shall discuss more closely in the next chapter §IV including the case ℏ is not a formal parameter. Let ζ_{1}, ζ̄_{1}, ζ_{2}, ζ̄_{2} be complex coordinates on C^{2}. On C^{∞}(C^{2})[[ℏ]], define an associative product ∗ by the Moyal product formula:

f ∗ g = f exp(−ℏ ∑_{i} \overleftarrow{∂_{ζ_{i}}} ∧^{·} \overrightarrow{∂ζ̄_{i}})g,

This algebra is called the formal Wick algebra. It is easy to see that

ρ = r^{2} = ζ̄_{1} ∗ ζ_{1} + ζ_{2} ∗ ζ̄_{2} = ζ̄_{1} · ζ_{1} + ζ_{2} · ζ̄_{2}.

Set C_{∗}^{2}= C^{2} − {0}. We define a one parameter group of automorphisms

R(e^{t}) : C^{∞}(C_{∗}^{2})[[ℏ]] → C^{∞}(C_{∗}^{2})[[ℏ]]

by setting on the generators as follows:

R(e^{t})ζ_{i} = e^{t}ζ_{i}, R(e^{t})ζ̄_{i} = e^{t}ζ̄_{i}, R(e^{t})ℏ = e^{2t}ℏ.

Define an closed subalgebra C ⊂ C^{∞}(C_{∗}^{2})[[ℏ]] by

C = {f ∈ C^{∞}(C_{∗}^{2})[[ℏ]]; R(e^{t})f = f}.

Since r^{2} is invertible, we set

μ = −2ℏr^{−2}, ξ_{1} = r^{−1} ∗ ζ_{1}, ξ_{2} = r^{−1} ∗ ζ_{2}, ξ_{1}^{∗} = ζ̄_{1} ∗ r^{−1}, ξ_{2}^{∗} = ζ̄_{2} ∗ r^{−1}.

We have

[μ^{−1}, ξ_{i}] = −ξ_{i}, [μ^{−1}, ξ_{i}^{∗}] = ξ_{i}^{∗} [ξ_{1}, ξ_{2}] = 0.
[ξ_{i}, ξ_{j}^{∗}]_{∗} = μ ∗ (δ_{ij} − ξ_{j}^{∗} ∗ ξ_{i}) for i, j = 1, 2, and
ξ_{1}^{∗} ∗ ξ_{1} + ξ_{2}^{∗} ∗ ξ_{2} = 1.

We see that C = C^{∞}(S^{3})[[μ]], and (C^{∞}(S^{3})[[μ]], ∗) is a μ-regulated algebra. This is an example of quantized contact algebra (cf. Definition 3.2).

Example 4

Let N be a C^{∞} riemannian manifold, and let T_{N}^{∗}, S_{N}^{∗} be the cotangent bundle, and its unit cosphere bundle respectively.

By using a local coordinate system (x^{1}, ⋯, x^{n}) on N, a point of the cotangent bundle is expressed in the form θ = ∑ y_{i}dx^{i}. We regard this as a 1-from on T_{N}^{∗}. Define a function ρ on T_{N}^{∗} by ρ = (∑ g^{ij}(x)y_{i}y_{j})^{1/2}.

It is known that C^{∞}(T_{N}^{∗}) with the standard Poisson bracket { , }_{s} is deformation quantizable, hence there is a ∗-product on C^{∞}(T_{N}^{∗})[[ℏ]].

Let Σ^{m}(T_{N}^{∗}) be the subspace consisting of functions which have the asymptotic expansion written in the form

ρ^{m}a_{m}(x; ω) + ⋯ + a_{0}(x; ω) + ρ^{−1}a_{1}(x; ω) + ⋯, a_{i} ∈ C^{∞}(S_{N}^{∗}).

By setting ℏ = 1, it is known that Σ^{0}(T_{N}^{∗}) is the space of symbols of classical pseudo-differential operators of order 0 on M, and then forms an algebra by the composition of operators (cf. [O]). If we denote the composition by ∗, then (Σ^{0}(T_{N}^{∗}), ∗) is an associative algebra. This is an algebra with following properties; For any a, b ∈ C^{∞}(S_{N}^{∗}), the product satisfies

a ∗ b ∼ a · b + \frac{1}{2ρ}\{a, b\}_{c} + ρ^{−2}π_{2}(a, b) + ⋯, π_{k}(a, b) ∈ C^{∞}(S_{N}^{∗}).

where π_{k}’s are bidifferential operators on S_{N}^{∗}.

Thus, setting μ = ρ^{−1}, we see that (C, ∗, B, μ) = (Σ^{0}(T_{N}^{∗}), ∗, C^{∞}(S_{N}^{∗}), μ) is an μ-regulated algebra by defining the topology so that (A.3) is satisfied.

Set Σ^{−∞}(T_{N}^{∗}) = ∩_{k}ρ^{−k}Σ^{0}(T_{N}^{∗}). Then (Σ^{0}(T_{N}^{∗})/Σ^{−∞}(T_{N}^{∗}), ∗, C^{∞}(S_{N}^{∗}), μ) is a formal μ-regulated algebra, and we see also

Σ^{0}(T_{N}^{∗})/Σ^{−∞}(T_{N}^{∗}) = C^{∞}(S_{N}^{∗})[[μ]].

§III.2 Deformation quantization

The examples given in the last section has the feature that B is the algebra of C^{∞} functions on some manifold M. Since we concern geometrical structures, we assume in the axioms of μ-regulated algebras that

B = C^{∞}(M) for some C^{∞} manifold M.  (3.4)

Theorem 3.1. (Localization theorem) For a μ-regulated algebra A with (3.4), we can change the filtration B[[μ]] so that ξ_{k}, π_{k} are local operators; i.e. ξ_{k} is a differential operator, and π_{k} is a bidifferential operator (cf. [OMY2]).

By localization theorem, ∗-product on A can be defined on the space C^{∞}(U)[[μ]] for every open subset U of M by the same product formula

f ∗ g = fg + μ ∗ π_{1}(f, g) + μ^{2}π_{2}(f, g) + ⋯

Since ad(μ^{−1}) is a derivation of (C, ∗), ξ_{0} is a derivation of (B, ·), Since [a, b] is a biderivation of C × A into A, the skew part π_{1}^{−} of π_{1} is a skew biderivation of B × B into B. Denote {f, g} = 2π_{1}^{−}(f, g). Since ad(μ^{−1}) is a derivation of A,

ξ_{0}({f, g}) = {ξ_{0}(f), g} + {f, ξ_{0}(g)}  (3.5)

Since μ^{−1}A forms a Lie algebra, we see that

{f, g}_{c} = fξ_{0}(g) − gξ_{0}(f) − {f, g}  (3.6)

makes B = C^{∞}(M) a Lie algebra.

Definition 3.2. On C^{∞}(M), suppose we have a derivation ξ_{0} and a skew biderivation { , }. We call that (C^{∞}(M), ·, { , }) is a

(a) Jacobi algebra ⇔ iff it satisfies (3.5) and (3.6) (cf. [L])

(b) Poisson algebra ⇔ ξ_{0} = 0 and Jacobi algebra

(c) symplectic algebra ⇔ a Poisson algebra, the dimension of the underlying manifold M is even, e.g. 2m, and the rank of Poisson bracket is full, i.e. 2m.

(d) contact algebra ⇔ ξ_{0} nowhere vanish, the dimension of the underlying manifold M is odd, e.g. 2m+1, and the rank of { , } is maximal everywhere, i.e. 2m (cf. [L]).

The space C^{∞}(M) on a contact manifold forms a contact algebra, and the space C^{∞}(M) on a symplectic manifold forms a symplectic algebra.

Now, we have the following quantization problem:

Do such algebras defined by classical term come from μ-regulated algebras?

If an algebra B (e.g. a Poisson algebra) is obtained as the first 2-components of a μ-regulated algebra A, then B is called to be deformation quantizable. A is called a quantized algebra. (e.g. quantized symplectic algebra, or qsymplectic algebra.)

The following is the list of partial answers for quantization problem

(1) Every Poisson algebra of constant rank is deformation quantizable (cf. [O]).

(2) In particular, every symplectic algebra is deformation quantizable (cf. [DL], [OMY1], [Fe]).

(3) Every contact algebra is deformation quantizable.([OMMY])

(4) Every Jacobi algebra is embedded in a Poisson algebra. Hence, the quantization problem of Jacobi algebra is reduced to the quantization problem of Poisson algebra. (5) On R^{n}, if Poisson bracket is given in the form {x_{i}, x_{j}} = c_{ij} + ∑ C_{ij}^{k}x_{k} then this Poisson algebra is deformation quantizable. In particular, (𝔤^{∗}; { , }) is deformation quantizable (cf. [O])

Quantization problem of Poisson algebra where the rank of { , } is not constant is difficult in general. However, it is not known non-quantizable Poisson algebra.

§III.3 Local generator systems and Local generator transformations (coordinate transformations)

Let (C^{∞}(M)[[μ]], ∗) be a formal μ regulated algebra. By localization theorem, ∗-product is defined on C^{∞}(U)[[μ]] every open subset U of M where C^{∞}(U) has the C^{∞}-topology and C^{∞}(U)[[μ]] has the product topology regarding C^{∞}(U)[[μ]] ≅ ∏_{i=0}^{∞} C^{∞}(U).

For elements ξ_{1}, ⋯, ξ_{n} ∈ C^{∞}(U)[[μ]], we write

ξ_{i} = x_{i} + μ ∗ f_{i}^{1} + μ^{2} ∗ f_{i}^{2} + ⋯

Definition 3.3. (ξ_{1}, ⋯, ξ_{n}) is called a local generator system, if x_{1}, ⋯, x_{n} are coordinate functions of U.

If (ξ_{1}, ⋯, ξ_{n}) is local generator system, then this generates a dense subalgebra of C^{∞}(U)[[μ]]. For any open subset V ⊂ U, (ξ_{1}, ⋯, ξ_{n}) is also a local generator system of C^{∞}(V)[[μ]]. We now define the notion of “coordinate transformations” by the reexpression of elements by means of another system of generators.

Suppose ξ_{1}, ⋯, ξ_{n} be a local generator system of (C^{∞}(U)[[μ]], ∗), and η_{1}, ⋯, η_{n} be a local generator system of (C^{∞}(V)[[μ]], ∗). Then, the generator transformation on (C^{∞}(U ∩ V)[[μ]], ∗) means simply the expression of ξ_{i} by using η_{1}, ⋯, η_{n}.

In localized algebras, we have in general much freedom to select generators, since many elements of the algebra have inverse. By this reason, we can choose some special local generator systems for localized algebras. We give here some examples, which may be called canonical local generator systems (CLGS).

Let (C^{∞}(M)[[μ]], ∗) be a formal μ-regulated algebra.

1) If (C^{∞}(M), ·, π^{−}) is a symplectic algebra, then there exists an open covering {U_{α}}_{α} of M such that there is a local generator system x_{1}, ⋯, x_{n}, y_{1}, ⋯, y_{n} ∈ C^{∞}(U_{α})[[μ]] with [X_{i}, x_{j}] = [y_{i}, y_{j}] = 0, [x_{i}, y_{j}] = μδ_{ij}.

2) If (C^{∞}(M), ·, ξ_{0}, π^{−}) is a contact algebra, Then there exists an open covering {U_{α}}_{α} of M such that there is a local generator system τ, x_{1}, ⋯, x_{n}, y_{1}, ⋯, y_{n} ∈ C^{∞}(U_{α})[[μ]] with

\begin{matrix}[X_{i}, x_{j}] = [y_{i}, y_{j}] = 0, [x_{i}, y_{j}] = μδ_{ij} [τ, μ] = 2μ^{2} \\ [τ, x_{i}] = μx_{i}, [τ, y_{i}] = μy_{i} \end{matrix}

There are several notions which can be defined by using the notion of local coordinate transformations. Let (C^{∞}(M)[[μ]], ∗) be a formal μ-regulated algebra.

A) (C^{∞}(M)[[μ]], ∗) is called a q-cotangent bundle if there exists an open covering {V_{α}} of M such that 1) there exists a CLGS x_{α}^{1}, ⋯, x_{α}^{n}, y_{1}^{α}, ⋯, y_{n}^{α} ∈ C^{∞}(V_{α})[[μ]] such that

[x_{α}^{i}, x_{α}^{j}] = [y_{i}^{α}, y_{j}^{α}] = 0 [x_{α}^{i}, y_{j}^{α}] = μδ_{j}^{i}

2) On every V_{α} ∩ V_{β}, two generators are related by x_{β}^{i} = φ_{αβ}^{i}(x_{α}), y_{i}^{β} = ∑_{k}(dφ_{αβ}^{−1}) ∗ y_{k}^{α}.

As is known, for every T_{N}^{∗} there exists a q-cotangent bundle (C^{∞}(T_{N}^{∗})[[μ]], ∗).

B) (C^{∞}(M)^{C}[[μ]], ∗) is called a q-Kähler manifold, if there exists an involutive anti-automorphism f → f̄, and an open covering {V_{α}} of M such that

1) There exists z_{1}^{α}, ⋯, z_{n}^{α} ∈ C^{∞}(V_{α})^{C}[[μ]] such that [z_{i}^{α}, z_{j}^{α}] = 0, ([z_{i}^{α}, z̄_{j}^{α}]) = non-degenerate

2) On every V_{α} ∩ V_{β} ≠ ∅, two generators are related by holomorphic transformations, i.e. z_{i}^{β} = φ_{αβ}^{i}(z^{α}).

It is known in [Ka] the following:

Theorem 3.4. If M is a Kähler manifold, then there exists (C^{∞}(M)^{C}[[μ]], ∗) which is a q-Kähler manifold.

§IV.1. Berezin representation

In this chapter §IV, we give some applications of the use of noncommutative algebras.

Let D_{Ω}(S^{2}) be the group of all volume preserving C^{∞} diffeomorphisms on the standard 2-sphere S^{2}. This is an infinite dimensional Lie group, i.e. a strong ILH-Lie group with the Lie algebra Γ_{Ω}(T_{S^{2}}) of all divergence free vector fields. It is easy to see that the Lie algebra Γ_{Ω}(T_{S^{2}}) is isomorphic to (C_{R}^{∞}(S^{2})/R, { , }): the space of all C^{∞} potential functions with the usual Poisson bracket { , }. It is known that Γ_{Ω}(T_{S^{2}}) is the space of all derivations of (C_{R}^{∞}(S^{2}), ·, { , }).

Similarly, it is known that D_{Ω}(S^{2}) is the group of all automorphisms of

(C_{R}^{∞}(S^{2}), ·, { , }).

In §VI, we shall show the following:

• There exists a quantum version of the Poisson algebra (C_{C}^{∞}(S^{2}), ·, { , }) as a certain complete topological associative algebra (V, ∗) over the complex number C with a special element μ which is in the center.

• There exists a matrix representation of the algebra (V, ∗) such that −μ is represented as a diagonal matrix −μ̂ = diag{I_{1}, 2^{−1}I_{2}, ⋯, k^{−1}I_{k}, ⋯ } where I_{k} is the identity matrix of rank k.

• (μ^{−1} ∗V, [ , ]) forms a Lie algebra and there exists also its matrix representation as a projective limit of finite dimensional Lie algebras of matrices, since every elements commute with μ^{−1}.

By this, we can exponentiate μ^{−1}a and we can make a group G̃ generated by exp μ^{−1} ∗ V, which is also a projective limit of finite dimensional Lie groups. This can be understood as an infinite dimensional Lie group with the Lie algebra μ^{−1} ∗ V with the projective limit topology from the system of finite dimensional Lie algebras of matrices. However this topology for μ^{−1} ∗V is much weaker than the original topology for μ^{−1} ∗ V.

It is easy to see that the Lie algebra of all derivations Der(V, ∗) is isomorphic to (ad(μ^{−1}∗V), [ , ]). It follows that the adjoint group Ad(G̃) acts as automorphisms of (V, ∗). This can be viewed also as an infinite dimensional Lie group with the Lie algebra Der(V, ∗) under the projective limit topology of finite dimensional Lie groups. It is known that G̃ can not be an infinite dimensional Lie group having the Lie algebra μ^{−1} ∗ V with the original topology.

On the other hand, we can see that every f ∈ V is asymptotically written in the form

f ∼ f_{0} + μ ∗ f_{1} + ⋯ + μ^{k} ∗ f_{k} + ⋯, f_{i} ∈ C_{C}^{∞}(S^{2}).

Let μ^{−1}∗V_{R} be the Lie subalgebra of all μ^{−1} ∗f such that the first component f_{0} is a real valued function, i.e. f_{0} ∈ C_{R}^{∞}(S^{2}). We give μ^{−1}∗V_{R} the restricted topology from the original topology of μ^{−1} ∗ V. Then, we see :

• There exists an infinite dimensional Lie group G with Lie algebra μ^{−1}∗V_{R}. Strictly speaking this is a regular Frechét Lie group in the sense [O].

• G is continuously embedded in G̃. It is clear that Ad(G) is a group consisting of automorphisms of (V, ∗), i.e. Ad(G) ⊂ Aut(V, ∗).

• There exists a continuous homomorphism from G onto the group D_{Ω}(S^{2}). Its kernel is the group generated by subalgebra with {f_{0} = 0}. Since D_{Ω}(S^{2}) has no closed finite codimensional normal subgroup, the above homomorphism cannot extend to G̃.

It is remarkable that the quantized objects are much easier to make a matrix representation.

§IV.2. Extension of Wick algebra.

To make clear the topology for μ^{−1} ∗ V, we start with the Wick algebra. Consider now an associative algebra W over the complex number field C generated by {ℏ, ζ_{1}, ζ̄_{1}, ζ_{2}, ζ̄_{2}} with the relations:

[ζ_{i}, ζ̄_{j}] = −2ℏδ_{ij}, [ζ_{i}, ζ_{j}] = [ζ̄_{i}, ζ̄_{j}] = 0, ℏ ∈ center  (4.1)

This is called the Wick algebra, where ℏ is a parameter moving in positive reals ℏ ∈ R_{+}. (Cf. Example 3 in §III.) W has a canonical involutive antiautomorphism a → ā. We denote its product by ∗. It is easy to see that

ζ̄_{1} ∗ ζ_{1} + ζ_{2} ∗ ζ̄_{2} = ζ_{1} ∗ ζ̄_{1} + ζ̄_{2} ∗ ζ_{2}(= ρ).

Let ζ_{1}, ζ̄_{1}, ζ_{2}, ζ̄_{2} be complex coordinates on C^{2}. Let C[ζ, ζ̄, ℏ] be the space of all polynomials on C^{2}. The Wick algebra W is linearly isomorphic to C[ζ, ζ̄, ℏ] and the product ∗ is given by the Moyal product formula given in §III. Example 3.

First, we have to take the topological completion of W: Let Σ^{0}(C^{2} ×R_{+}): ∀ C^{∞} functions on C^{2} × R_{+} with the asymptotic expansion

f ∼ f_{0} + f_{1} · ρ^{−1} + ⋯ + f_{k} · ρ^{−k} + ⋯, f_{k}(p, ℏ) ∈ C^{∞}(S^{3} × R_{+})

where ρ^{−k} is \frac{1}{ρ^{k}} in the ordinary product ·.

Let Σ^{−m}(C^{2} × R_{+}) be the subspace such that the asymptotic expansion start with ρ^{−m} where m ∈ Z. We set

Σ^{−∞}(C^{2} × R_{+}) = ⋂ Σ^{−m}(C^{2} × R_{+}),
Σ^{∞}(C^{2} × R_{+}) = ⋃ Σ^{m}(C^{2} × R_{+}).

Topology for Σ^{0}(C^{2} × R_{+}) is given as follows; For every m ∈ Z_{+}, set

Σ^{0}(C^{2}×R_{+}) = ∑_{k=0}^{m}⊕C^{∞}(S^{3}×R_{+})ρ^{−k} ⊕ Σ^{−(m+1)}(C^{2}×R_{+}).

Give the product topology of the uniform C^{∞} topology for each space. Denote this topology by T_{m}, and give the projective limit topology to Σ^{0}(C^{2}× R_{+}).

For Σ^{m}(C^{2} × R_{+}) (m > 0), we topologize by identifying this with Σ^{0}(C^{2}×R_{+})ρ^{m}. ∗-product extends to Σ^{∞}(C^{2}×R_{+}) by the following integral formula: Identifying C^{2} with R^{4}, ∗-product extended to Σ^{∞}(C^{2} × R_{+}) by the following oscillatory integral formula:

\begin{matrix}f ∗ g = & os-∫ f(x + X, y + ℏY)g(x + X′, y + ℏY′) \\ × e^{i(XY′−Y X′)} d̄X d̄Y d̄X′ d̄Y′, \end{matrix}  (4.2)

where x = (x_{1}, x_{2}), XY′ − Y X′ = ∑(X_{i}Y′_{i} − Y_{i}X′_{i}), dX =dX_{1}dX_{2} etc., and we compute this integral combining Taylor expansion of f and g up to an appropriate order.

Clearly, there exists a linear isomorphism:

C^{∞}(S^{3} × R_{+})[[ρ^{−1}]] ≅ Σ^{0}(C^{2} × R_{+})/Σ^{−∞}(C^{2} × R_{+}).

(Σ^{0}(C^{2} × R_{+}), ∗) defines an associative product ∗ on the factor space C^{∞}(S^{3} × R_{+})[[ρ^{−1}]].

On the other hand, since ℏ is not treated as a formal parameter, it is not obvious the existence of \sqrt{ρ} or \sqrt{ρ}^{−1} in (Σ^{∞}(C^{2} × R_{+}), ∗). To ensure this we compute first the ∗-exponential e_{∗}^{−t \frac{1}{2} ρ} as in §I. Suppose for a while that F_{t}(ρ) = e_{∗}^{−t \frac{1}{2} ρ}, and F_{t} is a function of one variable in s. By the product formula, we have

\begin{matrix}\frac{∂}{∂t}F_{t}(ρ) = −\frac{1}{2}ρ ∗ F_{t}(ρ) \\ = −\frac{1}{2}ρ · F_{t}(ρ) + ℏ^{2}F′_{t}(ρ) + ℏ^{2} \frac{1}{2}ρ · F′′_{t}(ρ). \end{matrix}

Hence, F_{t} must satisfy the same differential equation

\frac{∂}{∂t}F_{t}(s) = −\frac{1}{2}sF_{t}(s) + ℏ^{2}(F′_{t}(s) + \frac{1}{2}sF′′_{t}(s))

as in Lemma 2.3 with the initial condition F_{0}(s) = 1. Hence, we have

Lemma 4.1. The ∗-exponential e_{∗}^{− \frac{1}{2} ρ} is given by

e_{∗}^{− \frac{t}{2} ρ} = \frac{4e^{ℏt}}{(e^{ℏt} + 1)^{2}} exp\{−\frac{ρ}{ℏ}tanh\frac{ℏt}{2}\}

In particular, e_{∗}^{− \frac{t}{2} ρ} ∈ Σ^{−∞}(C^{2} ×R_{+}), for every t > 0 and lim_{t→∞} e_{∗}^{− \frac{t}{2} ρ} = 0, but

lim_{t→∞} e_{∗}^{− \frac{t}{2} (ζ_{1}∗ζ̄_{1}+ζ_{2}∗ζ̄_{2})} = lim_{t→∞} e_{∗}^{− \frac{t}{2} ρ}e^{ℏt} = 4e^{− \frac{ρ}{ℏ}}

We denote 4e^{− \frac{ρ}{ℏ}} by ϖ because this plays the same role ϖ in §1 and §2. As in Lemma 1.5, we have

Corollary 4.2. There exists (\frac{1}{2}ρ−z)^{−1} ∈ (Σ^{0}(C^{2}×R_{+}), ∗) for every Re z < ℏ. In particular, ρ is invertible.

We denote the inverse by ρ_{∗}^{−1}. Using the formula of Laplace transform, we define;

\sqrt[∗]{ρ_{∗}^{−1}} = \frac{1}{\sqrt{π}} ∫_{0}^{∞}\sqrt{t}^{−1}e_{∗}^{−tρ}dt.

We denote r_{∗}^{−1} = \sqrt[∗]{ρ_{∗}^{−1}}. It is easy to see (r_{∗}^{−1})^{2} = ρ_{∗}^{−1} and r_{∗}^{−1} ∈ Σ^{−1}(C^{2} × R_{+}). We define r_{∗} by r_{∗} = r_{∗}^{−1} ∗ ρ.

On the other hand by the product formula, we see that

ζ̄_{i} ∗ ϖ = 0 = ϖ ∗ ζ_{i}, i = 1, 2.

This implies that ϖ plays the role of vacuum and hence, we can use the subspace

Σ^{∞}(C^{2} × R_{+}) ∗ ϖ = C^{∞}(R_{+}) ⊗ C[ζ_{1}, ζ_{2}] ∗ ϖ

as the regular representation space of the algebra: That is, for every f we define the operator f̂ by f̂(a ∗ ϖ) = f ∗ a ∗ ϖ.

Remark. If ℏ is treated as a formal parameter, then ϖ is treated as 0. That is, ϖ = 0 in (C^{∞}(C^{2})[[ℏ]], ∗).

Every w ∈ Σ^{∞}(C^{2} × R_{+}) expressed by the following blockwise matrix:

ŵ = [ \begin{matrix}B_{1,1} & B_{1,2} & · & ⋯ & ⋯ & ⋯ \\ B_{2,1} & B_{2,2} & B_{2,3} & · & ⋯ & ⋯ \\ · & B_{3,2} & B_{3,3} & B_{3,4} & · & ⋯ \\ ⋯ & · & B_{4,3} & B_{4,4} & B_{4,5} & · \\ ⋯ & ⋯ & · & B_{5,4} & B_{5,5} & B_{5,6} \\ ⋮ & ⋮ & ⋮ & ... & ... & ... \end{matrix} ] ,

where B_{i,j} : i × j-matrix such that B_{i,j} = 0 for |i − j| ≫ 1. In fact, the generators of W satisfies B_{i,j} = 0 for |i − j| ≥ 2 and have the following form:

For ζ_{1}, B_{s+1,s} = \sqrt{2ℏ} [ \begin{matrix}\sqrt{s} & · & · & · \\ ⋮ & ⋮ & ... \\ · & · & \sqrt{2} \\ & & 0 & \sqrt{1} \\ 0, & ⋯ & ⋯ & 0 \end{matrix} ] other blocks are 0

For ζ_{2}, B_{s+1,s} = \sqrt{2ℏ} [ \begin{matrix}0 & ⋯ & ⋯ & 0 \\ \sqrt{1} & · & · & · \\ · & \sqrt{2} & · & · \\ ⋮ & ⋮ & ... \\ √ & \overline{s} \end{matrix} ] other blocks are 0

We also have ζ̄_{i} = ^{t}ζ_{i}. We see that the matrix expression of ϖ = 2e^{− \frac{ρ}{ℏ}}, and r are

ϖ̂ = diag{1, 0, 0, ⋯, 0, ⋯ }
r = diag{B_{1,1}, B_{2,2}, ⋯ B_{k,k} ⋯ },

where

B_{k,k} = \sqrt{2ℏ} [ \begin{matrix}√ & \overline{k} \\ √ & \overline{k} \\ & & . & . & . \\ & & & & & . & . & . \\ √ & \overline{k} \end{matrix} ] .

We construct matrix representations as subalgebras of the above matrix algebra. A matrix is blockwise diagonal, if it has the blockwise form such that B_{i,j} = 0 for i ≠ j. Since r is represented as above, every element which commutes with r is represented by blockwise diagonal matrices such that each block has a finite rank.

§IV.3. Noncommutative 3-sphere and reduction

In the ordinary mechanics, we take here a Hamiltonian H = r^{2}/2 and the energy surface H = c. Here instead of this, we consider a one parameter group of automorphisms R(e^{t}) of Σ^{∞}(C^{2} × R_{+}) by setting on the generators as follows (see Example 3):

R(e^{t})ζ_{i} = e^{t}ζ_{i}, R(e^{t})ζ̄_{i} = e^{t}ζ̄_{i}, R(e^{t})ℏ = e^{2t}ℏ.

Denote C: the closed subalgebra of Σ^{0}(C^{2} × R_{+}) consisting of all R(e^{t})- invariant elements.

By the existence of r_{∗}^{−1} in Σ^{∞}(C^{2} × R_{+}), we can see C is topologically generated by

\begin{aligned}μ = −2ℏr_{∗}^{−2}, & ξ_{1} = r_{∗}^{−1} ∗ ζ_{1}, ξ_{2} = r_{∗}^{−1} ∗ ζ_{2}, \\ & ξ̄_{1} = ζ̄_{1} ∗ r_{∗}^{−1}, ξ̄_{2} = ζ̄_{2} ∗ r_{∗}^{−1}. \end{aligned}

Precisely, these generate dense subalgebra in C. Since μ^{−1} ∗ ϖ = −ϖ, ξ̄_{i} ∗ ϖ = 0, we have

C ∗ ϖ = C[ξ_{1}, ξ_{2}] ∗ ϖ.

This can be used as the regular representation space of C. i.e. for every f ∈ C, we define the operator f̂ by f̂(a ∗ ϖ) = f ∗ a ∗ ϖ:

f̂: C[ξ_{1}, ξ_{2}] ∗ ϖ → C[ξ_{1}, ξ_{2}] ∗ ϖ.

We see that −μ̂^{−1} = diag{I_{1}, ⋯, kI_{k}, ⋯ } where I_{k} is the k × k identity matrix, and ξ̂_{i}, ˆ ξ̄_{i} are easily obtained.

Lemma 4.3. μ, ξ_{i}, ξ̄_{i} are in Σ^{0}(C^{2} × R_{+}) and these have the following relations:

ξ̄_{1} ∗ ξ_{1} + ξ̄_{2} ∗ ξ_{2} = 1, ξ_{1} ∗ ξ̄_{1} + ξ_{2} ∗ ξ̄_{2} = 1 + μ
[μ^{−1}, ξ_{i}] = −ξ_{i}, [μ^{−1}, ξ̄_{i}] = ξ̄_{i},
[ξ_{1}, ξ_{2}] = [ξ̄_{1}, ξ̄_{2}] = 0, [ξ_{i}, ξ̄_{j}] = μ(δ_{ij} − ξ̄_{j} ∗ ξ_{i})
for i, j = 1, 2.

Let C^{−∞} = C ⋂ Σ^{−∞}(C^{2} × R_{+}). Then, we see

C/C^{−∞} ≅ C^{∞}(S^{3})[[μ]] linearly.

Thus, (C, ∗) defines an associative product ∗ on the space C^{∞}(S^{3})[[μ]]. In what follows we shall explain that (C^{∞}(S^{3})[[μ]], ∗) can be viewed as a quantization of the contact algebra C^{∞}(S^{3}).

Next one gives the property of (C, ∗).

Theorem 4.4. By setting B = C^{∞}(S^{3}), C is a μ-regulated algebra.

The algebra C is called the non-commutative contact algebra on S^{3}.

This is because the following: Set for ∀a, b ∈ B

a ∗ b = ∑_{k≥0}μ^{k} ∗ π_{k}(a, b), π_{k}(a, b) ∈ B.  (α)

We see π_{0}(a, b) = a · b: usual commutative product, π_{1}^{−}; the skew part of π_{1}, gives a biderivation of B × B into B. On the other hand, ad(μ^{−1}) gives a derivation of C, where

[μ^{−1}, a] = −μ^{−1} ∗ [μ, a] ∗ μ^{−1}.
ad(μ^{−1})(a) = ξ_{0}(a) + ⋯ + μ^{k} ∗ ξ_{k}(a) + · · · .  (β)

\sqrt{−1}ξ_{0} is a derivation of (B, ·), hence a vector field, called a characteristic vector field, which gives an S^{1}-free action. (B, ·, \sqrt{−1}ξ_{0}, π_{1}^{−}) defines the ordinary contact structure on S^{3}.

It is easy to see that

[μ^{−1}∗C, C] ⊂ C, [μ^{−1}∗C, μ^{−1}∗C] ⊂ μ^{−1}∗C

Consider now the subalgebra of Σ^{0}(C^{2} × R_{+})

V = {f ∈ C; [μ, f] = 0}.

As μ plays the role of Hamiltonian, V can be viewed as the algebra obtained by the reduction procedure. It is easy to see ϖ ∗ V = V ∗ ϖ = Cϖ.

By Lemma 4.2, we see

[μ^{−1}, ξ_{i}] = −ξ_{i}, [μ^{−1}, ξ̄_{i}] = ξ̄_{i}

It follows that V is generated (topologically) by

μ, ξ_{1}∗ξ̄_{1}, ξ_{1}∗ξ̄_{2}, ξ_{2}∗ξ̄_{1} (= (ξ_{1} ∗ ξ̄_{2})^{−}).

Every element of V is represented as blockwise diagonal matrices.

Set V^{−∞} = V ⋂ Σ^{−∞}(C^{2} × R_{+}). Then, we see

V/V^{−∞} ≅ C^{∞}(S^{2})[[μ]].

Thus, (V, ∗) defines a noncommutative associative product on C^{∞}(S^{2})[[μ]].

(C^{∞}(S^{2})[[μ]], ∗) is the algebra obtained by the deformation quantization of the Poisson algebra (C^{∞}(S^{2}), ·, { , }). Note that the Lie algebra (C^{∞}(S^{2}), { , }) is the central extension of the Lie algebra of all volume preserving diffeomorphisms on S^{2}.

The Lie algebra (μ^{−1} ∗ V, [ , ]) can be viewed as a quantization of the Poisson algebra (C^{∞}(S^{2}), { , }).

If we set

H = ξ_{1}∗ξ̄_{1} − \frac{1+μ}{2}, Z = ξ_{1}∗ξ̄_{2}, Z^{∗} = ξ_{2}∗ξ̄_{1}

the above lemma shows that the algebra generated above, is an enveloping algebra of the Lie algebra sl_{μ}(2; C):

[H, Z] = −μ ∗ Z, [H, Z^{∗}] = μ ∗ Z^{∗}, [Z, Z^{∗}] = −2μ ∗ H

constrained by

(H + \frac{μ}{2})^{2} + Z ∗ Z^{∗} = \frac{1}{4}.

Note that (H + \frac{μ}{2})^{2} + Z ∗ Z^{∗} is in the center of the enveloping algebra. The matrix representations for H and Z are given as follows:

\begin{aligned}H & = diag\{B_{1,1}, B_{2,2}, ⋯, B_{k,k}, ⋯ \}, \\ Z & = diag\{B′_{1,1}, B′_{2,2}, ⋯, B′_{k,k}, ⋯ \}, \end{aligned}

with Z^{∗} = ^{t}Z. Here we set

B_{k,k} = \frac{1}{2k} diag\{k−1, k−3, ⋯, −(k−3), −(k−1)\},

B′_{k,k} = k^{−1} [ \begin{matrix}0, & \sqrt{(k−1)1}, \\ & 0, & \sqrt{(k−2)2}, \\ & & ... & ... \\ & & & 0, & \sqrt{1(k−1)} \\ & & & & 0 \end{matrix} ]

§IV.4. Local generators

By same computations as in §I, we see that for i = 1, 2,

e_{∗}^{− \frac{t}{2} ζ_{i}ζ̄_{i}} = \frac{2e^{ℏt/2}}{e^{ℏt} + 1} exp\{−\frac{ζ_{i}ζ̄_{i}}{ℏ} tanh\frac{ℏt}{2}\}, e_{∗}^{− \frac{t}{2} ζ_{i}ζ̄_{i}} ∈ Σ_{[i]}^{−∞}(C × R_{+}).

We see also that e_{∗}^{− \frac{t}{2} ζ̄_{i}∗ζ_{i}} = e_{∗}^{− \frac{t}{2} ζ_{i}ζ̄_{i}}e^{− \frac{t}{2} ℏ}, and

lim_{t→∞} e_{∗}^{− \frac{t}{2} ζ_{i}∗ζ̄_{i}} = lim_{t→∞} e_{∗}^{− \frac{t}{2} ζ_{i}ζ̄_{i}}e^{\frac{t}{2} ℏ} = 2e^{− \frac{ζ_{i}ζ̄_{i}}{ℏ}} .

It follows that ζ_{i}ζ̄_{i} and ζ̄_{i} ∗ ζ_{i} are invertible but ζ_{i} ∗ ζ̄_{i} are not invertible.

It follows that ζ_{i} has a left inverse (ζ̄_{i} ∗ζ_{i})^{−1} ∗ζ̄_{i}, but setting ϖ_{i} = 2e^{− \frac{ζ_{i}ζ̄_{i}}{ℏ}}, ϖ_{i} ∈ Σ_{[i]}^{−∞}(C × R_{+}),

ζ_{i} ∗ ((ζ̄_{i} ∗ ζ_{i})^{−1} ∗ ζ̄_{i}) = 1 − ϖ_{i}.

We have following formulas:

ϖ_{1} ∗ ϖ_{2} = ϖ_{2} ∗ ϖ_{1} = ϖ, ϖ_{i} ∗ ϖ_{i} = ϖ_{i},
ζ̄_{i} ∗ ϖ_{i} = 0 = ϖ_{i} ∗ ζ_{i}
ζ_{i} ∗ ϖ_{j} = ϖ_{j} ∗ ζ_{i} for i ≠ j
ϖ_{i} ∗ μ = μ ∗ ϖ_{i}, ϖ_{i} ∗ r_{∗} = r_{∗} ∗ ϖ_{i}.

Using the formula of Laplace transform, we define:

\sqrt[∗]{ζ̄_{i} ∗ ζ_{i}}^{−1} = \frac{1}{\sqrt{π}} ∫_{0}^{∞}\sqrt{t}^{−1}e_{∗}^{−tζ̄_{i}∗ζ_{i}}dt.

We have the following lemma:

Lemma 4.5. ζ_{i} ∗ \sqrt[∗]{ζ̄_{i} ∗ ζ_{i}}^{−1} ∈ Σ_{[i]}^{0}(C × R_{+}), but this is not the unitary part of ζ_{i}.

Set τ_{i} = e_{∗}^{− \frac{1}{2} ζ̄_{i}∗ζ_{i}}, and T_{i} = ζ_{i} ∗ \sqrt[∗]{ζ̄_{i} ∗ ζ_{i}}^{−1}, T_{i}^{∗} = \sqrt[∗]{ζ̄_{i} ∗ ζ_{i}}^{−1} ∗ ζ̄_{i}.

Lemma 4.6. We have the following formula: T_{i}^{∗}∗T_{i} = 1, T_{i}∗T_{i}^{∗} = 1−ϖ_{i}, τ_{i} ∗ T_{i} = e^{−ℏ}T_{i} ∗ τ_{i}, [T_{i}, T_{j}] = 0, [T_{i}, T_{j}^{∗}] = 0 (i ≠ j).

T_{i}, T_{i}^{∗} generate Töplitz algebra. We see easily that

T_{i}^{∗} ∗ ϖ_{i} = 0 = ϖ_{i} ∗ T_{i}.

Define for i = 1, 2

Σ̃_{[i]}^{0}(C^{2} × R_{+}) = algebra generated by {Σ^{0}(C^{2} × R_{+}), Σ_{[i]}^{0}(C × R_{+})}

C̃_{[i]}, Ṽ_{[i]} are defined similarly.

Localization of Σ^{0}(C^{2}×R_{+}), C or V means to embed it into other algebras, or more generally a homomorphism into another algebra.

We see that there exists (ξ̄_{1} ∗ξ_{1})^{−1} ∈ C̃_{[1]}, and there exists (1−ξ_{1} ∗ξ̄_{1})^{−1} ∈ C̃_{[2]}.

Define z ∈ Ṽ_{[1]}, w ∈ Ṽ_{[2]} by

\begin{matrix}z = ξ_{2} ∗ \{(ξ̄_{1} ∗ ξ_{1})^{−1} ∗ ξ̄_{1}\} = ζ_{2} ∗ \{(ζ̄_{1} ∗ ζ_{1})^{−1} ∗ ζ̄_{1}\}, \\ w = ξ_{1} ∗ \{(ξ̄_{2} ∗ ξ_{2})^{−1} ∗ ξ̄_{2}\} = ζ_{1} ∗ \{(ζ̄_{2} ∗ ζ_{2})^{−1} ∗ ζ̄_{2}\} \end{matrix}

These play the role of z = ξ_{2} ∗ ξ_{1}^{−1}, w = ξ_{1} ∗ ξ_{2}^{−1}. It is easy to obtain z ∗ ϖ_{1} = ϖ_{2} ∗ z = 0, w ∗ ϖ_{2} = ϖ_{1} ∗ w = 0, and ϖ = ϖ_{1} ∗ ϖ_{2} = ϖ_{2} ∗ ϖ_{1}.

Lemma 4.7. 1 + z ∗ z̄, 1 + z̄ ∗ z are invertible in Ṽ_{[1]}.

\begin{matrix}(1 + z ∗ z̄)^{−1} = ξ_{1} ∗ ξ̄_{1} − μ ∈ V \\ (1 + z̄ ∗ z)^{−1} = ξ_{1} ∗ ξ̄_{1} + ϖ_{1} ∈ Ṽ_{[1]} \end{matrix}

Proof. 1 + z ∗ z̄ = r^{2} ∗ (ζ̄_{1} ∗ ζ_{1})^{−1}. (1 + z ∗ z̄)^{−1} exists and = ξ_{1} ∗ ξ̄_{1} − μ. Since ∃(1 + z ∗ z̄)^{−1}, we see 1 − z̄∗(1 + z∗z̄)^{−1}∗z is the inverse of 1 + z̄ ∗ z. □

Note that ξ_{2} = z ∗ ξ_{1}, ξ_{1} = w ∗ ξ_{2}, ϖ_{1} ∗ ξ_{1} = 0, ϖ_{2} ∗ ξ_{2} = 0.

We see that {μ, z, z̄, ξ_{1}, ξ̄_{1}} topologically generates C̃_{[1]} and similarly {μ, w, w̄, ξ_{2}, ξ̄_{2}} topologically generates C̃_{[2]}.

[z∗z̄, z̄∗z] = 0, [w∗w̄, w̄∗w] = 0,
[z, ξ_{1}] ∗ (1 − ϖ_{1}) = 0, [w, ξ_{2}] ∗ (1 − ϖ_{2}) = 0
[z, ξ_{2}] = 0, [w, ξ_{1}] = 0,
ϖ_{1} ∗ (1 + z ∗ z̄) = −μ^{−1} ∗ ϖ_{1},
ϖ_{1} ∗ (1 + z̄ ∗ z) = ϖ_{1}.

z ∗ w = 1 − ϖ_{2} ∈ Ṽ_{[2]}, w ∗ z = 1 − ϖ_{1} ∈ Ṽ_{[1]}.

Using these, we have

\begin{matrix}[z, z̄] = μ(1 + z ∗ z̄) ∗ (1 + z̄ ∗ z) + μ^{−1} ∗ ϖ_{1} \\ [w, w̄] = μ(1 + w ∗ w̄) ∗ (1 + w̄ ∗ w) + μ^{−1} ∗ ϖ_{2} \end{matrix}

Comparing these with §II, (2.2), (2.3), we see that additional terms μ^{−1}ϖ_{i} appear by the reason that ℏ is treated as a positive variable.

We see that V_{[1]} is generated by {μ, z, z̄} ⊂ C̃_{[1]}, and Ṽ_{[2]} is generated by {μ, w, w̄} ⊂ C̃_{[2]}, and we see V = Ṽ_{[1]} ∩ Ṽ_{[2]}.

We have also

[z, z̄ ∗ (1 + z ∗ z̄)^{−1}] = −(μ + ϖ_{1}).

By these formulas, we see that the localized algebras can be represented as matrices; for k + l = m,

[ẑ, {μ ∗ ∂_{z}}ˆ ] = {μ + ϖ_{1}}ˆ

μ̂(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = −\frac{1}{m + 1} ∗ ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ.

ẑ(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = ξ_{1}^{k−1} ∗ ξ_{2}^{l+1} ∗ ϖ, ξ_{1}^{−1} = ξ_{2}^{m+1} = 0

ŵ(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = ξ_{1}^{k+1} ∗ ξ_{2}^{l−1} ∗ ϖ, ξ_{2}^{−1} = ξ_{1}^{m+1} = 0.

ϖ̂_{1}(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = δ_{k,0}ξ_{2}^{l} ∗ ϖ
ϖ̂_{2}(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = δ_{0,l}ξ_{1}^{k} ∗ ϖ

ϖ̂(ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ) = δ_{k,0}δ_{0,l} ∗ ϖ

We have also

ξ̄_{1} ∗ (1 + z̄ ∗ z) ∗ ξ_{1} = 1, ξ̄_{1} ∗ (1 + z̄ ∗ z) = (ξ̄_{1} ∗ ξ_{1})^{−1} ∗ ξ̄_{1},
ϖ_{1} ∗ (1 + z̄ ∗ z)^{−1} = −μ ∗ ϖ_{1}, ϖ_{1} ∗ (1 + z ∗ z̄)^{−1} = ϖ_{1}

w = ξ_{1} ∗ (1 − ξ_{1} ∗ ξ̄_{1})^{−1} ∗ ξ̄_{1} ∗ z̄ = f(μ, z, z̄).

Since [z, ξ_{1}] ∗ (1 − ϖ_{1}) = 0, [z, ξ_{2}] = 0, we see that ξ_{2}^{k} = (z ∗ ξ_{1})^{k} = z^{k} ∗ ξ_{1}^{k} (resp. ξ_{1}^{m} = w^{m} ∗ ξ_{2}^{m}), z^{k} ∗ ξ_{1}^{m} ∗ ϖ = 0, for k > m. Hence we see that ξ_{1}^{k} ∗ ξ_{2}^{l} ∗ ϖ, (k + l = m) can be viewed as a linear basis of the space H_{m} of all holomorphic sections of the holomorphic line bundle L^{m}. We see easily that H_{m} = {0} for m < 0 and H_{m} = P_{m} for m ≥ 0.

On Ṽ_{[1]} (resp. Ṽ_{[2]}), we see that

ξ_{1}^{k} ∗ ξ_{2}^{l} = z^{l} ∗ ξ_{1}^{m}, (resp. = w^{k} ∗ ξ_{2}^{m}).

The coordinate transformation is given by

ξ_{1}^{k} ∗ ξ_{2}^{l} = w^{k} ∗ ξ_{2}^{m} = (w^{k} ∗ ξ_{1}^{m}) ∗ z^{m} = z^{l} ∗ ξ_{1}^{m}.

Note these are well-defined:

T̃_{1} = ξ_{1} ∗ (ξ̄_{1} ∗ ξ_{1})^{−1/2}, T̃_{2} = ξ_{2} ∗ (ξ̄_{2} ∗ ξ_{2})^{−1/2}

{μ, z, z̄, T̃_{1}, T̃_{1}^{∗}}, {μ, w, w̄, T̃_{2}, T̃_{2}^{∗}} are local generator systems without constraint relations. The coordinate transformation may be understood as

{μ, w, w̄, T̃_{2}} = {μ, z^{−1}, z̄^{−1}, e_{∗}^{iθ_{+}} ∗T̃_{1}}

where e_{∗}^{iθ_{+}} is the unitary part of the polar decomposition of z. The existence of such a unitary part is given in C/{ϖ_{1}, ϖ_{2}}. By using z = ξ_{2} ∗{(ζ̄_{1} ∗ζ_{1})^{−1} ∗ ζ̄_{1} ∗ r_{∗}}, it is not hard to see that

e_{∗}^{iθ_{+}} ∗ T̃_{1} = T̃_{1} ∗ e_{∗}^{iθ_{+}} = T̃_{2}

The above coordinate transformation may be understood as that of quantum version of Hopf fibration of S^{3} over the Riemann sphere P^{1}(C) = S^{2}.

By a direct calculation, we have for k + l = m

\frac{1}{\sqrt{2ℏ}^{m}} \frac{1}{\sqrt{k!l!}}ζ_{1}^{k} ∗ ζ_{2}^{l} = \sqrt{−μ}^{−m} ∏_{j=0}^{m−1} \sqrt{1 + jμ} ∗ \frac{1}{\sqrt{k!l!}}ξ_{1}^{k} ∗ ξ_{2}^{l}.

Since μ̂^{−1} = −(m + 1) on Γ (L^{m}), we have on Ṽ_{[1]};

\frac{1}{\sqrt{2ℏ}^{m}} \frac{1}{\sqrt{k!l!}}ζ_{1}^{k} ∗ ζ_{2}^{l} = \sqrt{\frac{(m+1)!}{k!l!}}ξ_{1}^{k} ∗ ξ_{2}^{l} = \sqrt{\frac{(m+1)!}{k!l!}}z^{l} ∗ ξ_{1}^{m}

A linear basis of the space H_{m} of all holomorphic sections is given for k + l = m by

\begin{matrix}√ & \overline{(m+1)!} \\ √ & \overline{k!l!} \end{matrix}ξ_{1}^{k} ∗ ξ_{2}^{l} = \{ \begin{matrix}^{\begin{matrix}√ & \overline{(m+1)!} \\ √ & \overline{k!l!} \end{matrix}}z^{l} ∗ ξ_{1}^{m} (on Ṽ_{[1]}) \\ ^{\begin{matrix}√ & \overline{(m+1)!} \\ √ & \overline{k!l!} \end{matrix}}w^{k} ∗ ξ_{2}^{m} (on Ṽ_{[2]}) \end{matrix}

These form an orthonormal basis.

Thus the matrix representations for the above generators are written for each non-negative integer m as

(ξ_{1}∗ξ̄_{1})ˆ : \sqrt{\frac{(m+1)!}{(m−l)!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → \frac{m−l}{m+1} \sqrt{\frac{(m+1)!}{(m−l)!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ,
(ξ_{1}∗ξ̄_{2})ˆ : \sqrt{\frac{(m+1)!}{k!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → ^{\frac{\sqrt{(k+1)l}}{m+1}} \sqrt{\frac{(m+1)!}{(k+1)!(l−1)!}}z^{l−1}∗ξ_{1}^{m} ∗ ϖ,
(ξ_{2}∗ξ̄_{1})ˆ : \sqrt{\frac{(m+1)!}{k!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → ^{\frac{\sqrt{k(l+1)}}{m+1}} \sqrt{\frac{(m+1)!}{(k−1)!(l+1)!}}z^{l+1} ∗ ξ_{1}^{m} ∗ ϖ,

where we set z^{−1} = z^{m+1} = 0.

Here, note that {ξ̄_{1} ∗ξ_{1}, ξ̄_{2} ∗ξ_{1}, ξ̄_{1} ∗ξ_{2}} also forms a system of generators. To give the Berezin representation, it is convenient to use

{ξ̄_{1} ∗ ξ_{1}, ξ̄_{2} ∗ ξ_{1}, ξ̄_{1} ∗ ξ_{2}},

rather than {ξ_{1} ∗ ξ̄_{1}, ξ_{1} ∗ ξ̄_{2}, ξ_{2} ∗ ξ̄_{1}}.

We have easily:

\begin{matrix}ξ̄_{1} ∗ ξ_{1} = −\frac{μ}{2ℏ} ∗ \frac{1}{1 − μ} ∗ ζ̄_{1} ∗ ζ_{1} \\ ξ̄_{2} ∗ ξ_{1} = −\frac{μ}{2ℏ} ∗ \frac{1}{1 − μ} ∗ ζ̄_{2} ∗ ζ_{1}. \end{matrix}

These have also matrix representations as follows:

(ξ̄_{1} ∗ ξ_{1})ˆ : \sqrt{\frac{(m+1)!}{(m−l)!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → \frac{m+1−l}{m+2} \sqrt{\frac{(m+1)!}{(m−l)!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ,
(ξ̄_{2} ∗ ξ_{1})ˆ : \sqrt{\frac{(m+1)!}{k!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → ^{\frac{\sqrt{(k+1)l}}{m+2}} \sqrt{\frac{(m+1)!}{(k+1)!(l−1)!}}z^{l−1} ∗ ξ_{1}^{m} ∗ ϖ,
(ξ̄_{1} ∗ ξ_{2})ˆ : \sqrt{\frac{(m+1)!}{k!l!}}z^{l} ∗ ξ_{1}^{m} ∗ ϖ → ^{\frac{\sqrt{k(l+1)}}{m+2}} \sqrt{\frac{(m+1)!}{(k−1)!(l+1)!}}z^{l+1} ∗ ξ_{1}^{m} ∗ ϖ,

where we set z^{−1} = z^{m+1} = 0.

We now define integral operators to obtain representation: The following is easily obtained:

Lemma 4.8. Let z and v denote the complex variables on C. Then, for every integer m(≥ 0) the mapping

I_{m}(p)(z) = \frac{m+1}{π} ∫_{C} p(v)\frac{(1+zv̄)^{m}}{(1+vv̄)^{m}} \frac{1}{(1+vv̄)^{2}} dvdv̄

is the identity on the space P_{m} of all polynomials of degree up to m.

Since \frac{1}{(1+vv̄)^{2}} dvdv̄ is the volume form on S^{2}, the right hand of the equality in Lemma 4.8 can be viewed as the integral over S^{2}. Using Lemma 4.8, we define the projection operator P_{m} by

(P_{m}f)(z) = \frac{m+1}{π} ∫_{C} f(v, v̄)\frac{(1+zv̄)^{m}}{(1+vv̄)^{m}} \frac{1}{(1+vv̄)^{2}} dvdv̄

Define B(a)f for ∀f = ∑_{m} f_{m} ∗ ξ_{1}^{m}, f_{m} ∈ P_{m} by

B(a)f = ∑_{m≥0}P_{m}(a · f_{m}) ∗ ξ_{1}^{m}

for any a ∈ C^{∞}(S^{2}).

It is easy to see that B(a) defines a linear operator of ∑ ⊕P_{m} ∗ ξ_{1}^{m} ∗ ϖ into itself. By a direct computation:

ξ̄_{1} ∗ ξ_{1} =B(\frac{1}{1+z̄z}), ξ̄_{2} ∗ ξ_{1} =B(\frac{z̄}{1+z̄z}), ξ̄_{1} ∗ ξ_{2} =B(\frac{z}{1+z̄z})

This constitutes the Berezin representation. Using these, we have an operator representation of the algebra V. This coincides with the matrix representation given in the beginning. B(a)f can be expressed as an integral operator

B(a)f = _{∑m≥0}\{\frac{m+1}{π} ∫_{C} a(v, v̄)f_{m}(v)\frac{(1+zv̄)^{m}}{(1+vv̄)^{m}} \frac{1}{(1+vv̄)^{2}} dvdv̄\} ∗ ξ_{1}^{m}.

We see that any element of the algebra C are represented as matrices, although these are not blockwise diagonal matrices.

Hence, we have the following:

Theorem 4.9. The matrix representation extends to the algebra C. The algebra V is represented by blockwise diagonal matrices, and μ^{−1} is represented as a diagonal matrix. Thus, we have a matrix representation for the Lie algebra μ^{−1}∗C.

In particular, the Lie algebra μ^{−1}∗V is represented by blockwise diagonal matrices. Since each block is finite rank, we see that the group G generated by exp μ^{−1} ∗ V is also represented by blockwise diagonal matrices. This means that μ^{−1} ∗ V is continuously embedded in a projective limit of finite dimensional Lie algebras.

It follows also that G has a series of finite codimensional normal subgroups N_{k} such that

N_{k} ⊃ N_{k+1}, ∩N_{k} = {e}

Thus, there is an isomorphism from G into the group F obtained as the projective limit of finite dimensional Lie groups.

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Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba, 278, Japan

Received July 27, 1999