ANALYTIC CENTRAL EXTENSIONS OF INFINITE DIMENSIONAL WHITE NOISE ∗–LIE ALGEBRAS

LUIGI ACCARDI AND ANDREAS BOUKAS

Abstract. The connection between the ∗–Lie algebra of the Renormalized Higher Powers of White Noise (RHPWN) and the centerless Virasoro (or Witt)-Zamolodchikov-w_{∞} ∗–Lie algebra of conformal field theory, as well as the associated Fock space construction, have recently been established ([?]–[?]). In this paper we describe a method for looking for a special class of central extensions of the RHPWN and w_{∞} ∗–Lie algebras called ”analytic”, i.e. central extensions where the defining cocycles can be written as formal power series of the indices of the RHPWN and w_{∞} generators. Our method is also applied to the well known Virasoro central extension of the Witt algebra.

1. Introduction: The theory of the RHPWN

Quantum white noise is defined by a pair of mutually adjoint operator valued distributions a_{t}^{†} (creation density) and a_{t} (annihilation density) (i.e. (a_{s})^{∗} = a_{s}^{†}) satisfying the Boson commutation relations

[a_{t}, a_{s}^{†}] = δ(t − s) ; [a_{t}^{†}, a_{s}^{†}] = [a_{t}, a_{s}] = 0

where t, s ∈ ℝ and δ is the Dirac delta function and, here and in the following, [x, y] = xy−yx denotes the commutator. The attempt to give a meaning to the higher powers of white noise led to the following commutation relations which are ill defined because of the presence of the “powers” of the δ–function:

(1.1)  [a_{t}^{†}^{n}a_{t}^{k}, a_{s}^{†}^{N}a_{s}^{K}] =

ϵ_{k,0} ϵ_{N,0} ∑_{L≥1} \binom{k}{L}N^{(L)} a_{t}^{†}^{n} a_{s}^{†}^{N−L} a_{t}^{k−L} a_{s}^{K} δ^{L}(t − s)

−ϵ_{K,0} ϵ_{n,0} ∑_{L≥1} \binom{K}{L}n^{(L)} a_{s}^{†}^{N} a_{t}^{†}^{n−L} a_{s}^{K−L} a_{t}^{k} δ^{L}(t − s)

where for n, k ∈ {0, 1, 2, ...} we have used the notation ϵ_{n,k} = 1−δ_{n,k}, where δ_{n,k} is Kronecker’s delta, and x^{(y)} = x(x − 1) ⋯ (x − y + 1) with x^{(0)} = 1.

Date: January 10, 2012.

The problem of giving an interesting and useful meaning to these commutation relations has been widely studied in physics and is called the renormalization problem. The quantum probabilistic approach to this problem consists of three steps:

(i) find a renormalization procedure which gives a meaning to the higher powers of the δ–function in such a way that, after renormalization, the commutation relations (??) still define a ∗–Lie algebra (ii) construct a unitary representation for this Lie algebra (typically the Fock one) (iii) determine the statistics canonically associated to this representation

In the past years the realization of this program has revealed some deep and unexpected connections with the theory of independent increment, stationary, additive stochastic processes, with conformal field theory and with string theory. The fact that in some cases the existence of a unitary representation is not guaranteed for the Lie algebra itself, but only for a central extension of it, leads to the analysis of central extensions of the Lie algebras obtained with the given renormalization procedure. In the present paper we will investigate the central extensions of the Renormalized Higher Powers of White Noise (RHPWN) ∗–Lie algebra and the closely related Virasoro-Zamolodchikov-w_{∞} ∗–Lie algebra.

The one mode RHPWN Lie algebra is the infinite dimensional ∗–Lie algebra with generators (B_{k}^{n}) (n, k ∈ ℕ), commutation relations

(1.2)  [B_{k}^{n}, B_{K}^{N}]_{RHP W N} = (k N − K n) B_{k+K−1}^{n+N−1}

and involution

(B_{k}^{n})^{∗} = B_{n}^{k}

The renormalization procedure which has led to this algebra is described in [?] and [?].

The Virasoro-Zamolodchikov-w_{∞} ∗–Lie algebra (cf. [?]) is the infinite dimensional ∗–Lie algebra, with generators (B̂_{k}^{n}) (n ∈ ℕ, n ≥ 2, k ∈ ℤ), commutation relations

(1.3)  [B̂_{k}^{n}, B̂_{K}^{N}]_{w_{∞}} = (k (N − 1) − K (n − 1)) B̂_{k+K}^{n+N−2}

and involution

\binom{}{B̂_{k}^{n}}^{∗} = B̂_{−k}^{n}

ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS 3

In [?] we proved that the two algebras (more precisely their closure in a suitable topology) coincide in the sense that the generators of the w_{∞} algebra can be expressed as suitable convergent series of the generators of the RHPWN algebra and vice versa. The ∗–Lie sub– algebra of the w_{∞} algebra, generated by the family (B̂_{k}^{2}) (k ∈ ℤ) is the centerless Virasoro (or Witt) algebra which is known to admit a non trivial central extension (see Appendix below). The central extensions of the w_{∞} algebra have been widely studied in the physical literature. In particular, Bakas (cf. [?], see also [?]) proved that the w_{∞} ∗–Lie algebra does not admit non-trivial central extensions. That was done by showing that, after a suitable contraction which yields the w_{∞} commutation relations, the central terms appearing in the algebra W_{∞}, which is defined as the N → ∞ limit of the finite–dimensional Zamolodchikov Lie algebras W_{N}, vanish.In what follows, after recalling in section (??) some standard facts about central extensions of Lie algebras, in section (??) we introduce a new more direct method to attack this problem and we apply this method to the RHPWN algebra. In section (??) we apply our method to provide a new proof of the triviality of the analytic central extensions of w_{∞} which, being direct, avoids the need to introduce the approximating algebras W_{∞} and their contractions. Finally in the Appendix, section (??), we apply our method to give a new proof of the existence of a unique non trivial central extension of the Witt–Virasoro algebra. As in the above quoted papers in the physical literature, we assume the possibility to expand in power series the cocycles of the extensions considered: for this reason we speak of analytical central extensions. We conjecture the possibility of dispensing of the analyticity condition by means of a purely algebraic approach to the problem. This development is the subject of a paper in preparation. However the method used here may be of independent interest for those cases in which a purely algebraic proof might not be available.

2. Central extensions of Lie algebras

We begin by recalling some standard facts about central extensions of Lie algebras (cf. [?]). If L and L̃ are two complex Lie algebras, we say that L̃ is a one-dimensional central extension of L with central element E if there is a Lie algebra exact sequence

0 ↦ ℂ E ↦ L̃ ↦ L ↦ 0

where ℂ E is the one-dimensional trivial Lie algebra and the image of ℂ E is contained in the center of L̃:

(2.1)  [l_{1}, E]L̃ = 0 ,  ∀l_{1} ∈ L

where [·, ·]L̃ are the Lie brackets in L̃. For ∗–Lie algebras we also require that the central element E is self–adjoint

(2.2)  (E)^{∗} = E

A 2-cocycle on L is a bilinear form ϕ : L × L ↦ ℂ on L satisfying, for all l_{1}, l_{2} ∈ L, the skew-symmetry condition

(2.3)  ϕ(l_{1}, l_{2}) = −ϕ(l_{2}, l_{1})

(in particular ϕ(l, l) = 0 for all l ∈ L) and the 2-cocycle identity:

(2.4)  ϕ([l_{1}, l_{2}]_{L}, l_{3}) + ϕ([l_{2}, l_{3}]_{L}, l_{1}) + ϕ([l_{3}, l_{1}]_{L}, l_{2}) = 0

One-dimensional central extensions of L are classified by 2-cocycles in the sense that L̃ is a central extension of L if and only if, as vector space, it is the direct sum

(2.5)  L̃ = M ⊕ ℂ E

where M is a Lie algebra isomorphic to L, and there exists a 2-cocycle on L such that Lie brackets in L̃ are given by

(2.6)  [l_{1}, l_{2}]L̃ = [l_{1}, l_{2}]_{L} + ϕ(l_{1}, l_{2}) E

where [·, ·]_{L} are the Lie brackets in L. A central extension is trivial if the corresponding 2-cocycle ϕ is uniquely determined by a linear function f : L ↦ ℂ through the identity

ϕ(l_{1}, l_{2}) = f([l_{1}, l_{2}]_{L}) , ∀l_{1}, l_{2} ∈ L (2.7)

Such a 2-cocycle is called a 2-coboundary, or a trivial 2-cocycle. Two extensions are called equivalent if each of them is a trivial extension of the other. This is the case if and only if the difference of the corresponding 2-cocycles is a trivial cocycle. A central extension L̃ of L is called universal whenever there exists a homomorphism from L̃ to any other central extension of L. A Lie algebra L possesses a universal central extension if and only if L is perfect (i.e. L = [L, L]). In this case, the universal central extension of L is unique up to isomorphism.

Definition 1. Let L be a Lie algebra with generators

{l_{k}^{n} ; n, k ∈ ℤ}

and let L̃ be a central extension of L. If the 2-cocycle ϕ associated with L̃ has the form

ϕ(l_{k}^{n}, l_{K}^{N}) = ∑_{m,s,M,S≥0} a_{m,s,M,S} n^{m} k^{s} N^{M} K^{S} ; a_{m,s,M,S} ∈ ℂ

for some convergent power series in the variables (n, k, N, K), then we say that L̃ is an analytic central extension of L.

3. Analytic Central extensions of the 1–mode RHPWN ∗–Lie algebra

Throughout this section we assume that { \atop RHP W N} is a central extension of the 1–mode RHPWN ∗–Lie algebra and we denote by c(n, k; N, K) the restriction of the corresponding 2–cocycle to the 1–mode RHPWN generators (B_{k}^{n}). Thus, in the notation (??):

(3.1)  c(n, k; N, K) = ϕ(B_{k}^{n}, B_{K}^{N}) ∈ ℂ

and

(3.2)  [B_{k}^{n}, B_{K}^{N}]{ \atop RHP W N} = [B_{k}^{n}, B_{K}^{N}]_{RHP W N} + c(n, k; N, K) E
= (k N − K n) B_{k+K−1}^{n+N−1} + c(n, k; N, K) E

The skew-symmetry of ϕ and the adjointness condition (??) imply that for all nonnegative integers n, k, N, K

(3.3)  c(n, k; N, K) = −c(N, K; n, k)

and

(3.4)  c(n, k; N, K) = −\overline{c(k, n; K, N)} = \overline{c(K, N; k, n)}

In particular, for N = n and K = k (??) implies that

(3.5)  c(n, k; n, k) = 0

If at least one of the n, k, N, K is negative we set

(3.6)  c(n, k; N, K) = 0

Lemma 1. On the 1–mode RHPWN generators B_{k}^{n} the 2-cocycle identity (??) is equivalent to

(k_{1} n_{2} − k_{2} n_{1}) c(n_{1} + n_{2} − 1, k_{1} + k_{2} − 1; n_{3}, k_{3}) (3.7)

+(k_{2} n_{3} − k_{3} n_{2}) c(n_{2} + n_{3} − 1, k_{2} + k_{3} − 1; n_{1}, k_{1})
+(k_{3} n_{1} − k_{1} n_{3}) c(n_{3} + n_{1} − 1, k_{3} + k_{1} − 1; n_{2}, k_{2}) = 0

Proof. For all n_{i}, k_{i} ≥ 0, where i = 1, 2, 3, making use of (??) we have

0 = ϕ([B_{k_{1}}^{n_{1}}, B_{k_{2}}^{n_{2}}]_{RHP W N}, B_{k_{3}}^{n_{3}}) + ϕ([B_{k_{2}}^{n_{2}}, B_{k_{3}}^{n_{3}}]_{RHP W N}, B_{k_{1}}^{n_{1}}) + ϕ([B_{k_{3}}^{n_{3}}, B_{k_{1}}^{n_{1}}]_{RHP W N}, B_{k_{2}}^{n_{2}})

= (k_{1}n_{2}−k_{2}n_{1})ϕ(B_{k_{1}+k_{2}−1}^{n_{1}+n_{2}−1}, B_{k_{3}}^{n_{3}})+(k_{2}n_{3}−k_{3}n_{2})ϕ(B_{k_{2}+k_{3}−1}^{n_{2}+n_{3}−1}, B_{k_{1}}^{n_{1}})+(k_{3}n_{1}−k_{1}n_{3})ϕ(B_{k_{3}+k_{1}−1}^{n_{3}+n_{1}−1}, B_{k_{2}}^{n_{2}})+ = (k_{1}n_{2} −k_{2}n_{1})c(n_{1} +n_{2} −1, k_{1} +k_{2} −1, n_{3}, k_{3})+(k_{2}n_{3} −k_{3}n_{2})c(n_{2} +n_{3} −1, k_{2} +k_{3} −1, n_{1}, k_{1})

+(k_{3}n_{1} − k_{1}n_{3})c(n_{3} + n_{1} − 1, k_{3} + k_{1} − 1, n_{2}, k_{2})

□

Remark 1.

We notice that the sum of the first and third (resp. second and fourth) arguments in the three 2-cocycle values c(n_{2} + n_{3} − 1, k_{2} + k_{3} − 1; n_{1}, k_{1}), c(n_{1} + n_{2} − 1, k_{1} + k_{2} − 1; n_{3}, k_{3}) and c(n_{3} + n_{1} − 1, k_{3} + k_{1} − 1; n_{2}, k_{2}) appearing in (??) is equal to n_{1} + n_{2} + n_{3} − 1 (resp. k_{1} + k_{2} + k_{3} − 1).

Proposition 1. Let S, M ∈ {−1, 0, 1, ...} be given and let n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} be nonnegative integers satisfying

(3.8)  n_{1} + n_{2} + n_{3} = S + 1 ; k_{1} + k_{2} + k_{3} = M + 1

For i ∈ {1, 2, 3} let

(3.9)  ψ(n_{i}, k_{i}) = c(S − n_{i}, M − k_{i}; n_{i}, k_{i})

with the skew-symmetry condition (derived from (??), (??) and (??))

(3.10)  ψ(n_{i}, k_{i}) = −ψ(S − n_{i}, M − k_{i})

Then, all analytic (in the sense of definition ??) solutions ψ of ( the ψ–form of (??))

(3.11) (k_{2} n_{3} − k_{3} n_{2}) ψ(n_{1}, k_{1}) + (k_{3} n_{1} − k_{1} n_{3}) ψ(n_{2}, k_{2}) + (k_{1} n_{2} − k_{2} n_{1}) ψ(n_{3}, k_{3}) = 0

i.e all ψ satisfying (??) for all n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} satisfying (??), are of the form

(3.12)  ψ(n_{i}, k_{i}) = z (M n_{i} − S k_{i})

where z = z_{S,M} ∈ ℂ.

Note: The dependence of z on S and M is justified by the fact that, in accordance with Remark ?? above, all ψ’s appearing in (??) must have the same z.

Proof. In view of (??) and Definition ??, let

ψ(n_{i}, k_{i}) = ∑_{r,ρ≥0} a_{r,ρ} n_{i}^{r} k_{i}^{ρ}

be an analytic solution of (??). Then (??) becomes

a_{0,0} (k_{2} n_{3}−k_{3} n_{2}+k_{3} n_{1}−k_{1} n_{3}+k_{1} n_{2}−k_{2} n_{1})+a_{1,0}·0+a_{0,1}·0+ ∑_{r+ρ>1} a_{r,ρ} f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) = 0

where

f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) = (k_{2} n_{3} − k_{3} n_{2}) n_{1}^{r} k_{1}^{ρ} + (k_{3} n_{1} − k_{1} n_{3}) n_{2}^{r} k_{2}^{ρ} + (k_{1} n_{2} − k_{2} n_{1}) n_{3}^{r} k_{3}^{ρ}

We notice that, if M ≠ −1 then f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) ≢ 0 for all n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} satisfying (??). Since k_{2} n_{3} − k_{3} n_{2} + k_{3} n_{1} − k_{1} n_{3} + k_{1} n_{2} − k_{2} n_{1} ≢ 0 either, it follows that a_{0,1} and a_{1,0} are arbitrary complex numbers and a_{0,0} = a_{r,ρ} = 0 for r + ρ > 1. If M = −1 then k_{1} + k_{2} + k_{3} = 0 implies that k_{1} = k_{2} = k_{3} = 0 and so, by (??), ψ(n_{1}, k_{1}) = 0 which is of the form (??) with z = 0. Thus

(3.13)  ψ(n_{i}, k_{i}) = a_{1,0} n_{i} + a_{0,1} k_{i}

By (??)

ψ(n_{1}, k_{1}) = −ψ(S − n_{1}, M − k_{1})

which combined with (??) implies that

(3.14)  a_{1,0} S + a_{0,1} M = 0

Case (i). S ≠ 0 and M ≠ 0: Then (??) implies that a_{1,0} = −a_{0,1} \frac{M}{S} and letting a_{0,1} = −z S, z ∈ ℂ, we find that a_{1,0} = z M and (??) becomes

ψ(n_{i}, k_{i}) = z (M n_{i} − S k_{i})

Case (ii). S = 0 and M = 0: Then n_{1} + n_{2} + n_{3} = 1 and k_{1} + k_{2} + k_{3} = 1 and so one of the n_{i} and one of the k_{i} are equal to 1 and the rest are equal to 0. If (n_{1}, n_{2}, n_{3}) = (1, 0, 0) and (k_{1}, k_{2}, k_{3}) = (1, 0, 0) then

ψ(n_{1}, k_{1}) = c(n_{2} + n_{3} − 1, k_{2} + k_{3} − 1; n_{1}, k_{1}) = c(−1, −1; 1, 1) = 0 ( by (??))
ψ(n_{2}, k_{2}) = c(n_{3} + n_{1} − 1, k_{3} + k_{1} − 1; n_{2}, k_{2}) = c(0, 0; 0, 0) = 0 ( by (??))
ψ(n_{3}, k_{3}) = c(n_{1} + n_{2} − 1, k_{1} + k_{2} − 1; n_{3}, k_{2}) = c(0, 0; 0, 0) = 0 ( by (??))

Thus

ψ(n_{i}, k_{i}) = 0 = z (0 n_{i} − 0 k_{i})

in agreement with (??). The cases (n_{1}, n_{2}, n_{3}) = (0, 1, 0), (k_{1}, k_{2}, k_{3}) = (0, 1, 0) and (n_{1}, n_{2}, n_{3}) = (0, 0, 1), (k_{1}, k_{2}, k_{3}) = (0, 0, 1) are similar.

If (n_{1}, n_{2}, n_{3}) = (1, 0, 0) and (k_{1}, k_{2}, k_{3}) = (0, 1, 0) then

ψ(n_{1}, k_{1}) = c(n_{2} + n_{3} − 1, k_{2} + k_{3} − 1; n_{1}, k_{1}) = c(−1, 0; 1, 0) = 0 ( by (??))
ψ(n_{2}, k_{2}) = c(n_{3} + n_{1} − 1, k_{3} + k_{1} − 1; n_{2}, k_{2}) = c(0, −1; 0, 1) = 0 ( by (??))
ψ(n_{3}, k_{3}) = c(n_{1} + n_{2} − 1, k_{1} + k_{2} − 1; n_{3}, k_{2}) = c(0, 0; 0, 0) = 0 ( by (??))

Thus

ψ(n_{i}, k_{i}) = 0 = z (0 n_{i} − 0 k_{i})

in agreement with (??). All other cases n_{i_{0}} = 1, n_{i} ≠ 0, i ≠ i_{0}, and k_{j_{0}} = 1, n_{j} ≠ 0, j ≠ j_{0}, where i_{0} ≠ j_{0}, are similar.

Case (iii). S = 0 and M ≠ 0: Then, by (??), a_{0,1} = 0 and by (??)

ψ(n_{i}, k_{i}) = a_{1,0} n_{i} = z (M n_{i} − 0 k_{i})

in agreement with (??) for z = \frac{a_{1,0}}{M}.

Case (iv). S ≠ 0 and M = 0: Then, by (??), a_{1,0} = 0 and by (??)

ψ(n_{i}, k_{i}) = a_{0,1} k_{i} = z (0 n_{i} − S k_{i})

in agreement with (??) for z = −\frac{a_{0,1}}{S}. □

Corollary 1. Let n, k, N, K be nonnegative integers. Then

c(n, k; N, K) = z_{n+N,k+K} · (k N − K n)

where z_{n+N,k+K} ∈ ℝ.

Proof. By proposition ??

c(n_{1} + n_{2} − 1, k_{1} + k_{2} − 1; n_{3}, k_{3}) = z ((k_{1} + k_{2} − 1) n_{3} − (n_{1} + n_{2} − 1) k_{3})

which letting n_{1} + n_{2} − 1 = n, k_{1} + k_{2} − 1 = k, n_{3} = N and k_{3} = K implies that

c(n, k; N, K) = z (k N − n K)

where, by proposition ??, z = z_{n+N,k+K} ∈ ℂ. By (??), z = z̄ and so z_{n+N,k+K} ∈ ℝ.

□

Theorem 1. All analytic central extensions (??) of the 1–mode RHPWN ∗–Lie algebra are trivial.

Proof. Define a complex valued function f on the 1–mode RHPWN generators by

f(B_{k}^{n}) = z_{n+1,k+1}

where z_{n+1,k+1} is as in Corollary ??, and extend by linearity the definition of f to the whole RHPWN algebra. Then

f([B_{k}^{n}, B_{K}^{N}]_{RHP W N}) = f ((k N − K n) B_{k+K−1}^{n+N−1})
= (k N − K n) f (B_{k+K−1}^{n+N−1})
= (k N − K n) z_{n+N,k+K}
= c(n, k; N, K)
= ϕ(B_{k}^{n}, B_{K}^{N})

Thus the central extension is trivial.

□

4. Analytic central extensions of the 1–mode w_{∞} Lie algebra

Denote by w̃_{∞} an analytic central extension of the 1–mode w_{∞} Lie algebra such that on the w_{∞} generators

[B̂_{k}^{n}, B̂_{K}^{N}]w̃_{∞} = [B̂_{k}^{n}, B̂_{K}^{N}]_{w_{∞}} + c(n, k; N, K) E
= (k (N − 1) − K (n − 1)) B̂_{k+K}^{n+N−2} + c(n, k; N, K) E

where n, N ≥ 2, k, K ∈ ℤ and the coefficients c(n, k; N, K) are defined in terms of the 2–cocycle ϕ by:

c(n, k; N, K) = ϕ(B̂_{k}^{n}, B̂_{K}^{N}) ∈ ℂ

As in the 1–mode RHPWN case, for all integers n, N, k, K with n, N ≥ 2:

(4.1)  c(n, k; N, K) = −c(N, K; n, k)

and

(4.2)  c(n, k; N, K) = −\overline{c(n, −k; N, −K)}

ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS 9

For n < 2 and/or N < 2 we define

(4.3)  c(n, k; N, K) = 0

Lemma 2. On the 1–mode w_{∞} generators B̂_{k}^{n} equation (??) is equivalent to (k_{2} (n_{3} − 1) − k_{3} (n_{2} − 1)) c(n_{2} + n_{3} − 2, k_{2} + k_{3}; n_{1}, k_{1}) (4.4)

+(k_{3} (n_{1} − 1) − k_{1} (n_{3} − 1)) c(n_{3} + n_{1} − 2, k_{3} + k_{1}; n_{2}, k_{2})
+(k_{1} (n_{2} − 1) − k_{2} (n_{1} − 1)) c(n_{1} + n_{2} − 2, k_{1} + k_{2}; n_{3}, k_{3}) = 0

Proof. The proof is similar to that of lemma ?? □ Proposition 2. Let S ≥ 5 (thus removing the Virasoro sector S = 4) and M ∈ ℤ be given and let n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} ∈ ℤ satisfy n_{1}, n_{2}, n_{3} ≥ 2 and

(4.5)  n_{1} + n_{2} + n_{3} = S + 2 ; k_{1} + k_{2} + k_{3} = M

For i ∈ {1, 2, 3} let

ψ(n_{i}, k_{i}) = c(S − n_{i}, M − k_{i}; n_{i}, k_{i})

where by (??)

(4.6)  ψ(n_{i}, k_{i}) = −ψ(S − n_{i}, M − k_{i})

Then, all analytic (in the sense of definition ??) solutions ψ of ( the ψ–form of (??)) (k_{2} (n_{3} − 1) − k_{3} (n_{2} − 1)) ψ(n_{1}, k_{1}) + (k_{3} (n_{1} − 1) − k_{1} (n_{3} − 1)) ψ(n_{2}, k_{2}) (4.7)

+(k_{1} (n_{2} − 1) − k_{2} (n_{1} − 1)) ψ(n_{3}, k_{3}) = 0

i.e all ψ satisfying (??) for all n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} ∈ ℤ satisfying n_{1}, n_{2}, n_{3} ≥ 2 and (??), are of the form

(4.8)  ψ(n_{i}, k_{i}) = z (M (n_{i} − 1) − (S − 2) k_{i})

where z = z_{S,M} ∈ ℂ. Proof. Let

ψ(n_{i}, k_{i}) = ∑_{r,ρ≥0} a_{r,ρ} (n_{i} − 1)^{r} k_{i}^{ρ}

Then (??) becomes

a_{0,0} (k_{2} (n_{3} − 1) − k_{3} (n_{2} − 1) + k_{3} (n_{1} − 1) − k_{1} (n_{3} − 1) + k_{1} (n_{2} − 1) − k_{2} (n_{1} − 1))
+a_{1,0} · 0 + a_{0,1} · 0 + ∑_{r+ρ>1} a_{r,ρ} f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) = 0

where

f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) =
(k_{2} (n_{3} − 1) − k_{3} (n_{2} − 1)) (n_{1} − 1)^{r} k_{1}^{ρ} + (k_{3} (n_{1} − 1) − k_{1} (n_{3} − 1)) (n_{2} − 1)^{r} k_{2}^{ρ}
+(k_{1} (n_{2} − 1) − k_{2} (n_{1} − 1)) (n_{3} − 1)^{r} k_{3}^{ρ}

We notice that, since the case S = 4 has been excluded, f_{r,ρ}(n_{1}, n_{2}, n_{3}, k_{1}, k_{2}, k_{3}) ≢ 0 for all n_{1}, k_{1}, n_{2}, k_{2}, n_{3}, k_{3} satisfying (??). Since the same is true for the coefficient of a_{0,0}, it follows that a_{0,1} and a_{1,0} are arbitrary complex numbers and a_{0,0} = a_{r,ρ} = 0 for r + ρ > 1. Thus

(4.9)  ψ(n_{i}, k_{i}) = a_{1,0} (n_{i} − 1) + a_{0,1} k_{i}

which combined with (??) implies that

(4.10)  a_{1,0} (S − 2) + a_{0,1} M = 0

Case (i). M ≠ 0: Then (??) implies that a_{1,0} = −a_{0,1} \frac{M}{S−2} and letting a_{0,1} = −z (S − 2), z ∈ ℂ, we find that a_{1,0} = z M and (??) becomes

ψ(n_{i}, k_{i}) = z (M (n_{i} − 1) − (S − 2) k_{i})

Case (ii). M = 0: Then a_{1,0} (S − 2) + a_{0,1} · 0 = 0 which implies that a_{1,0} = 0 i.e.

ψ(n_{i}, k_{i}) = a_{0,1} k_{i} = z (0 (n_{i} − 1) − (S − 2) k_{i})

where z = \frac{a_{0,1}}{2−S} in agreement with (??).

□

Corollary 2. Let n, k, N, K be integers, with n, N ≥ 2 and n + N ≥ 5 (thus excluding the Virasoro sector n + N = 4). Then

c(n, k; N, K) = z_{n+N,k+K} · (k (N − 1) − K (n − 1))

where z_{n+N,k+K} ∈ ℝ.

Proof. By proposition ??

c(n_{1} + n_{2} − 2, k_{1} + k_{2}; n_{3}, k_{3}) = z ((k_{1} + k_{2}) (n_{3} − 1) − (n_{1} + n_{2} − 3) k_{3})

which letting n_{1} + n_{2} − 2 = n, k_{1} + k_{2} = k, n_{3} = N and k_{3} = K implies that

c(n, k; N, K) = z (k (N − 1) − (n − 1) K)

where, by proposition ??, z = z_{n+N,k+K} ∈ ℂ. By (??)

z (k (N − 1) − (n − 1) K) = −\overline{z ((−k) (N − 1) − (n − 1) (−K))} = z̄(k (N − 1) − (n − 1) K)

i.e. z = z̄ which implies that z_{n+N,k+K} ∈ ℝ.

□

Theorem 2. All analytic central extensions of the 1–mode w_{∞} ∗–Lie algebra (with its Virasoro sector removed) are trivial.

Proof. Define a complex valued function f on the 1–mode w_{∞} generators by

f(B̂_{k}^{n}) = z_{n+2,k}

where z_{n+2,k} is as in Corollary ??. Linearly extend the definition of f to all of w_{∞} . Then

f([B̂_{k}^{n}, B̂_{K}^{N}]_{w_{∞}}) = f ((k (N − 1) − K (n − 1)) B̂_{k+K}^{n+N−2})
= (k (N − 1) − K (n − 1)) f \binom{}{B̂_{k+K}^{n+N−2}}
= (k (N − 1) − K (n − 1)) z_{n+N,k+K}
= c(n, k; N, K)
= ϕ(B̂_{k}^{n}, B̂_{K}^{N})

Thus the central extension is trivial.

□

5. Appendix: Analytic central extensions of the Witt Lie algebra

It is well known (cf. [?]) that the (1–mode) Witt algebra W with generators {L_{m}; m ∈ ℤ} satisfying the commutation relations

(5.1)  [L_{m}, L_{n}]_{W} = (m − n) L_{m+n}

with adjointness condition

(L_{m})^{∗} = L_{−m}

can be centrally extended to the Virasoro algebra V through

(5.2)  [L_{m}, L_{n}]_{V} = (m − n) L_{m+n} + δ_{m+n,0} m (m^{2} − 1) E

The Witt algebra with commutation relations (??) is a sub-algebra of the w_{∞} Lie algebra with commutation relations (??).

In Proposition ?? below we describe how the technique used in the proof of proposition ?? can be used to derive (??). Our approach differs from the traditional one, where recursions are constructed so as to yield (??), in that we obtain all non-trivial analytic central extensions of the Witt algebra which we then show to be equivalent to (??).

So let W̃ be an analytic central extension of the Witt ∗–Lie algebra such that on the W generators

[L_{k}, L_{K}]W̃ = (k − K) L_{k+K} + c(2, k; 2, K) E

where k, K ∈ ℤ

c(2, k; 2, K) = ϕ(L_{k}, L_{K}) ∈ ℂ

where ϕ is a cocycle as in (??). As in the RHPWN case, for all integers k and K

(5.3)  c(2, k; 2, K) = −c(2, K; 2, k)

and

(5.4)  c(2, k; 2, K) = −\overline{c(2, −k; 2, −K)}

Proposition 3. The Virasoro central extension (??) of the Witt algebra (??) is non-trivial.

Proof. Suppose that (??) is a trivial extension of (??). Then there exists a linear complexvalued function f defined on the Witt algebra such that

(5.5)  f ([L_{m}, L_{n}]_{W}) = δ_{m+n,0} m (m^{2} − 1)

By (??) and the linearity of f, equation (??) yields

(m − n) f (L_{m+n}) = δ_{m+n,0} m (m^{2} − 1)

from which for n = 0 we obtain

m f (L_{m}) = δ_{m,0} m (m^{2} − 1)

which implies that

f(L_{m}) = \{ \begin{matrix}0 & if m ≠ 0 \\ z & if m = 0 \end{matrix}

where z ∈ ℂ is arbitrary. But then, for any m ≠ 0,

(5.6)  z = f(L_{0}) = f (\frac{1}{2 m} [L_{m}, L_{−m}]_{W}) = \frac{1}{2 m} f ([L_{m}, L_{−m}]_{W})
= \frac{1}{2 m} m (m^{2} − 1) = \frac{m^{2} − 1}{2}

and also

(5.7)  z = f(L_{0}) = f (\frac{1}{4 m} [L_{2 m}, L_{−2 m}]_{W}) = \frac{1}{4 m} f ([L_{2 m}, L_{−2 m}]_{W})
= \frac{1}{4 m} 2 m (4 m^{2} − 1) = \frac{4 m^{2} − 1}{2}

By (??) and (??) it follows that f(L_{0}) cannot be consistently defined. Thus such an f cannot exist, therefore the central extension (??) of (??) is non-trivial.

□

Lemma 3. On the Witt algebra generators L_{k} (= B̂_{k}^{2} in the notation of w_{∞}) equation (??) is equivalent to

(5.8) (k_{2}−k_{3}) c(2, k_{2}+k_{3}; 2, k_{1})+(k_{3}−k_{1}) c(2, k_{3}+k_{1}; 2, k_{2})+(k_{1}−k_{2}) c(2, k_{1}+k_{2}; 2, k_{3}) = 0

Proof. The proof is identical to that of lemma ?? □

Proposition 4. (i) The Witt algebra (??) analytic central extensions are given by

(5.9)  [L_{m}, L_{n}]W̃ = (m − n) L_{m+n} + δ_{m+n,0} m (β m^{2} + α) E

where α, β ∈ ℝ. (ii) The analytic central extensions (??) are trivial if and only if β = 0. (iii) All analytic central extensions of the form (??) are equivalent to the Virasoro central extension (??) which corresponds to the choice α = −β = −\frac{1}{12}.

Proof. (i) Letting

θ(k_{i}) = c(2, −k_{i}; 2, k_{i})

(??) becomes

(k_{2} − k_{3}) θ(k_{1}) + (k_{3} − k_{1}) θ(k_{2}) + (k_{1} − k_{2}) θ(k_{3}) = 0

which, assuming that

θ(k_{i}) = ∑_{ρ≥0} a_{ρ} k_{i}^{ρ}

yields that for all integers k_{1}, k_{2}, k_{3} such that k_{1} + k_{2} + k_{3} = 0

∑_{ρ≥0} a_{ρ} ((k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ}) = 0

Clearly for ρ = 0 and ρ = 1

(k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ} ≡ 0

For ρ = 3

(k_{2} − k_{3}) k_{1}^{3} + (k_{3} − k_{1}) k_{2}^{3} + (k_{1} − k_{2}) k_{3}^{3}
= k_{2} k_{1} (k_{1}^{2} − k_{2}^{2}) + k_{3} k_{2} (k_{2}^{2} − k_{3}^{2}) + k_{1} k_{3} (k_{3}^{2} − k_{1}^{2})
= k_{2} k_{1} (k_{1} + k_{2}) (k_{1} − k_{2}) + k_{3} k_{2} (k_{2} + k_{3}) (k_{2} − k_{3}) + k_{1} k_{3} (k_{3} + k_{1}) (k_{3} − k_{1})
= −k_{2} k_{1} k_{3} (k_{1} − k_{2}) − k_{3} k_{2} k_{1} (k_{2} − k_{3}) − k_{1} k_{3} k_{2} (k_{3} − k_{1})
= −k_{2} k_{1} k_{3} (k_{1} − k_{2} + k_{2} − k_{3} + k_{3} − k_{1}) ≡ 0

Thus a_{0}, a_{1}, a_{3} are arbitrary. For ρ ≠ 0, 1, 3

(k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ} ≢ 0

Thus all other a_{ρ}’s are equal to zero and so

θ(k_{i}) = a_{0} + a_{1} k_{i} + a_{3} k_{i}^{3}

By (??)

θ(k_{i}) = c(2, −k_{i}; 2, k_{i}) = −c(2, k_{i}; 2, −k_{i}) = −θ(−k_{i})

which implies that a_{0} = 0. Similarly, by (??)

θ(k_{i}) = c(2, −k_{i}; 2, k_{i}) = −\overline{c(2, k_{i}; 2, −k_{i})} = −\overline{θ(−k_{i})}

which implies that for all integers k_{i}

a_{1} k_{i} + a_{3} k_{i}^{3} = ā_{1} k_{i} + ā_{3} k_{i}^{3}

and so a_{1} = ā_{1} and a_{3} = ā_{3} i.e. a_{1} = β and a_{3} = α where α, β ∈ ℝ. We have therefore proved (??) for n + m = 0. To prove it for n + m ≠ 0 we let n_{1} = n_{2} = n_{3} = 2 in (??) and, assuming that k_{1} + k_{2} + k_{3} = L ≠ 0, we obtain

(k_{2} − k_{3}) c(2, L − k_{1}; 2, k_{1}) + (k_{3} − k_{1}) c(2, L − k_{2}; 2, k_{2}) + (k_{1} − k_{2}) c(2, L − k_{3}; 2, k_{3}) = 0

which letting

θ(k_{i}) = c(2, L − k_{i}; 2, k_{i})

becomes

(k_{2} − k_{3}) θ(k_{1}) + (k_{3} − k_{1}) θ(k_{2}) + (k_{1} − k_{2}) θ(k_{3}) = 0

which assuming that

θ(k_{i}) = ∑_{ρ≥0} a_{ρ} k_{i}^{ρ}

yields that for all integers k_{1}, k_{2}, k_{3} such that k_{1} + k_{2} + k_{3} = L

∑_{ρ≥0} a_{ρ} ((k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ}) = 0

As before, for ρ = 0 and ρ = 1

(k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ} ≡ 0

and so a_{0}, a_{1} are arbitrary. For ρ ≥ 2

(k_{2} − k_{3}) k_{1}^{ρ} + (k_{3} − k_{1}) k_{2}^{ρ} + (k_{1} − k_{2}) k_{3}^{ρ} ≢ 0

thus a_{ρ} = 0 for all ρ ≥ 2 and so

θ(k_{i}) = a_{0} + a_{1} k_{i}

By (??)

θ(k_{i}) = c(2, L − k_{i}; 2, k_{i}) = −c(2, k_{i}; 2, L − k_{i}) = −θ(L − k_{i})

implies that a_{0} + a_{1} k_{i} = −a_{0} − a_{1}(L − k_{i}), i.e. 2 a_{0} + a_{1} L = 0 and so, since L is arbitrary, a_{0} = a_{1} = 0. Thus θ(k_{i}) ≡ 0.

(ii) Let ϕ(L_{n}, L_{m}) = δ_{n+m,0} (α m + β m^{3}). If β = 0 then the linear extension of the function

f(L_{n}) = −δ_{n,0} \frac{α}{2}

to all of the Witt algebra satisfies f([L_{n}, L_{m}]_{W}) = ϕ(L_{n}, L_{m}) and so the extension is trivial. Conversely, if β ≠ 0 and f is a linear function satisfying f([L_{n}, L_{m}]_{W}) = ϕ(L_{n}, L_{m}) then

(n − m) f(L_{n+m}) = δ_{n+m,0} (α m + β m^{3})

which for n + m = 0, m ≠ 0 and n ≠ m implies that

f(L_{0}) = \frac{α + β m^{2}}{−2}

which means that f(L_{0}) cannot be consistently defined. Therefore such an f does not exist and so the extension is not trivial.

(iii) As pointed out already, the factor \frac{1}{12} in (??) is for traditional reasons only and can therefore be replaced by an arbitrary real number β ≠ 0. We may thus write (??) and (??) as

[L_{m}, L_{n}]_{V} = (m − n) L_{m+n} + δ_{m+n,0} (m^{3} − m) β E

and

[L_{m}, L_{n}]W̃ = (m − n) L_{m+n} + δ_{m+n,0} (m^{3} + \frac{α}{β} m) β E

respectively. Letting

ϕ(L_{n}, L_{m}) = δ_{n+m,0} (m^{3} + \frac{α}{β} m)

and

ψ(L_{n}, L_{m}) = δ_{n+m,0} (m^{3} − m)

we see that

ω(L_{n}, L_{m}) = (ϕ − ψ)(L_{n}, L_{m}) = δ_{n+m,0} (\frac{α}{β} + 1) m

is a trivial cocycle since

ω(L_{n}, L_{m}) = f([L_{n}, L_{m}]_{W})

where

f(L_{k}) = −δ_{k,0} \frac{\frac{α}{β} + 1}{2}

Thus ϕ and ψ are equivalent cocycles.

□

Remark 2.

From the purely algebraic point of view, i.e. forgetting about its white noise origin, the RHPWN ∗–Lie algebra (??) can be generalized to n, k, N, K ∈ ℤ. One then wonders if a Virasoro type extension exists for some sector of the generalized RHPWN ∗–Lie algebra. The natural thing to do is to consider generators of the form B_{k}^{1} where k ∈ ℤ (in analogy to the Witt algebra generators B̂_{k}^{2}). Taking n_{1} = n_{2} = n_{3} = 1 and M = −1 in (??) we obtain

(5.10)  (k_{2} − k_{3}) ψ(1, k_{1}) + (k_{3} − k_{1}) ψ(1, k_{2}) + (k_{1} − k_{2}) ψ(1, k_{3}) = 0

where ψ(1, k_{i}) = −ψ(1, −1 − k_{i}) and so, letting θ(k_{i}) = ψ(1, k_{i}) = ∑_{ρ≥0} a_{ρ} k_{i}^{ρ}, we have the skew-symmetry condition θ(k_{i}) = −θ(−1 − k_{i}). Substituting in (??) and expanding we find that

a_{0} · 0 + a_{1} · 0 + a_{3} · 0 + ∑_{ρ≠0,1,3} a_{ρ} ((k_{2} − k_{3})k_{1}^{ρ} + (k_{3} − k_{1})k_{2}^{ρ} + (k_{1} − k_{2})k_{3}^{ρ})

where in the summation on the right hand side, (k_{2} − k_{3})k_{1}^{ρ} + (k_{3} − k_{1})k_{2}^{ρ} + (k_{1} − k_{2})k_{3}^{ρ} ≢ 0. Thus for all ρ ≠ 0, 1, 3 we have that a_{ρ} ≡ 0 and so θ(k_{i}) = a_{0} + a_{1} k_{i} + a_{3} k_{i}^{3}. By the above skew-symmetry condition it follows that a_{3} = 0 and a_{1} = 2 a_{0}. Therefore, letting a_{1} = λ, we have that θ(k_{i}) = \frac{λ}{2} + λ k_{i} which is of the form (??) with z = −\frac{λ}{2}. Thus no non-trivial analytic Virasoro type extension of the generalized RHPWN ∗–Lie algebra can be obtained in this way.

References

[1] Accardi, L., Boukas, A.: Renormalized higher powers of white noise (RHPWN) and conformal field theory, Infinite Dimensional Analysis, Quantum Probability, and Related Topics 9, No. 3, pages 353-360 (2006) .

[2] : The emergence of the Virasoro and w_{∞} Lie algebras through the renormalized higher powers of quantum white noise , International Journal of Mathematics and Computer Science, 1, No.3, pages 315–342 (2006) .

[3] : Renormalized Higher Powers of White Noise and the Virasoro–Zamolodchikov–w_{∞} Algebra, Reports on Mathematical Physics, Volume 61, number 1, pages 1-11 (2008) .

[4] : Fock representation of the renormalized higher powers of white noise and the Virasoro– Zamolodchikov–w_{∞} ∗–Lie algebra, J. Phys. A: Math. Theor. 41, pages 1–12 (2008).

[5] : Lie algebras associated with the renormalized higher powers of white noise, Communications on Stochastic Analysis Vol. 1, No. 1, pages 57-69 (2007).

[6] Accardi, L., Boukas, A., Franz, U.: Renormalized powers of quantum white noise, Infinite Dimensional Analysis, Quantum Probability, and Related Topics, 9, No. 1, pages 129–147 (2006)

[7] Bakas, I.: The structure of the W_{inf} algebra, Commun. Math. Phys. 134, pages 487-508 (1990)

[8] Feinsilver, P. J., Schott, R.: Algebraic structures and operator calculus. Volumes I and III, Kluwer (1993)

[9] Fuchs, J., Schweigert C. : Symmetries, Lie Algebras and Representations (A graduate course for physicists), Cambridge Monographs on Mathematical Physics, Cambridge University Press (1997)

[10] Ketov, S. V.: Conformal field theory, World Scientific (1995)

[11] Wanglai L., Wilson R.: Central extensions of some Lie algebras, Proceedings of the American Mathematical Society, Vol. 126, No. 9, pages 2569–2577 (1998).

Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy

E-mail address: accardi@volterra.mat.uniroma2.it
URL: http://volterra.mat.uniroma2.it

Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi, Athens 15342, Greece

E-mail address: andreasboukas@acgmail.gr