AccBou(Analytic-CE).tex 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 ([1]–[5]). 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 at (annihilation density) (i.e. (as )∗ = a†s ) satisfying the Boson commutation relations [at , a†s ] = δ(t − s) ; [a†t , a†s ] = [at , as ] = 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)
n
N
[a†t akt , a†s aK s ] = µ ¶ X k n N −L k−L K L ²k,0 ²N,0 N (L) a†t a†s at as δ (t − s) L L≥1 X µK ¶ n−L K−L k L N −²K,0 ²n,0 n(L) a†s a†t as at δ (t − s) L L≥1
where for n, k ∈ {0, 1, 2, ...} we have used the notation ²n,k = 1−δn,k , where δn,k is Kronecker’s delta, and x(y) = x(x − 1) · · · (x − y + 1) with x(0) = 1. 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: Date: August 4, 2015. 1
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LUIGI ACCARDI AND ANDREAS BOUKAS
(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 (1.1) 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-Zamolodchikovw∞ ∗–Lie algebra. The one mode RHPWN Lie algebra is the infinite dimensional ∗–Lie algebra with generators (Bkn ) (n, k ∈ N), commutation relations (1.2)
n+N −1 N [Bkn , BK ]RHP W N = (k N − K n) Bk+K−1
and involution (Bkn )∗ = Bnk The renormalization procedure which has led to this algebra is described in [1] and [2]. The Virasoro-Zamolodchikov-w∞ ∗–Lie algebra (cf. [10]) is the infinite dimensional ∗–Lie ˆ n ) (n ∈ N, n ≥ 2, k ∈ Z), commutation relations algebra, with generators (B k (1.3) and involution
ˆ n, B ˆ N ]w∞ = (k (N − 1) − K (n − 1)) B ˆ n+N −2 [B k K k+K ³
ˆn B k
´∗
ˆn =B −k
In [3] 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. ˆ 2 ) (k ∈ Z) is the The ∗–Lie sub–algebra of the w∞ algebra, generated by the family (B k 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. [7], see also [10]) 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 WN , vanish.
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
3
In what follows, after recalling in section (2) some standard facts about central extensions of Lie algebras, in section (3) we introduce a new more direct method to attack this problem and we apply this method to the RHPWN algebra. In section (4) 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 (5), 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. [11]). e are two complex Lie algebras, we say that L e is a one-dimensional central extension If L and L of L with central element E if there is a Lie algebra exact sequence e 7→ L 7→ 0 0 7→ C E 7→ L where C E is the one-dimensional trivial Lie algebra and the image of C E is contained in e the center of L: (2.1)
[l1 , E]Le = 0
,
∀l1 ∈ L
e For ∗–Lie algebras we also require that the central where [·, ·]Le are the Lie brackets in L. element E is self–adjoint (2.2)
(E)∗ = E
A 2-cocycle on L is a bilinear form φ : L × L 7→ C on L satisfying, for all l1 , l2 ∈ L, the skew-symmetry condition (2.3)
φ(l1 , l2 ) = −φ(l2 , l1 )
(in particular φ(l, l) = 0 for all l ∈ L) and the 2-cocycle identity: (2.4)
φ([l1 , l2 ]L , l3 ) + φ([l2 , l3 ]L , l1 ) + φ([l3 , l1 ]L , l2 ) = 0
e is a One-dimensional central extensions of L are classified by 2-cocycles in the sense that L central extension of L if and only if, as vector space, it is the direct sum (2.5)
e = M ⊕ CE L
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LUIGI ACCARDI AND ANDREAS BOUKAS
where M is a Lie algebra isomorphic to L, and there exists a 2-cocycle on L such that Lie e are given by brackets in L (2.6)
[l1 , l2 ]Le = [l1 , l2 ]L + φ(l1 , l2 ) 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 7→ C through the identity (2.7)
φ(l1 , l2 ) = f ([l1 , l2 ]L )
,
∀l1 , l2 ∈ L
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 e if the difference of the corresponding 2-cocycles is a trivial cocycle. A central extension L e of L is called universal whenever there exists a homomorphism from L 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 {lkn ; n, k ∈ Z} e be a central extension of L. If the 2-cocycle φ associated with L e has the form and let L X N φ(lkn , lK )= am,s,M,S nm k s N M K S ; am,s,M,S ∈ C m,s,M,S≥0
e is an for some convergent power series in the variables (n, k, N, K), then we say that L analytic central extension of L. 3. Analytic Central extensions of the 1–mode RHPWN ∗–Lie algebra ^ Throughout this section we assume that 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 (Bkn ). Thus, in the notation (2.6): (3.1)
N c(n, k; N, K) = φ(Bkn , BK )∈C
and (3.2)
N N [Bkn , BK ]RHP = [Bkn , BK ]RHP W N + c(n, k; N, K) E ^ WN n+N −1 + c(n, k; N, K) E = (k N − K n) Bk+K−1
The skew-symmetry of φ and the adjointness condition (2.2) imply that for all nonnegative integers n, k, N, K (3.3) and
c(n, k; N, K) = −c(N, K; n, k)
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
(3.4)
5
c(n, k; N, K) = −c(k, n; K, N ) = c(K, N ; k, n)
In particular, for N = n and K = k (3.3) 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 Bkn the 2-cocycle identity (2.4) is equivalent to (3.7)
(k1 n2 − k2 n1 ) c(n1 + n2 − 1, k1 + k2 − 1; n3 , k3 ) +(k2 n3 − k3 n2 ) c(n2 + n3 − 1, k2 + k3 − 1; n1 , k1 ) +(k3 n1 − k1 n3 ) c(n3 + n1 − 1, k3 + k1 − 1; n2 , k2 ) = 0
Proof. For all ni , ki ≥ 0, where i = 1, 2, 3, making use of (3.3) we have 0 = φ([Bkn11 , Bkn22 ]RHP W N , Bkn33 ) + φ([Bkn22 , Bkn33 ]RHP W N , Bkn11 ) + φ([Bkn33 , Bkn11 ]RHP W N , Bkn22 ) +n2 −1 +n3 −1 +n1 −1 = (k1 n2 −k2 n1 )φ(Bkn11+k , Bkn33 )+(k2 n3 −k3 n2 )φ(Bkn22+k , Bkn11 )+(k3 n1 −k1 n3 )φ(Bkn33+k , Bkn22 )+ 2 −1 3 −1 1 −1 = (k1 n2 −k2 n1 )c(n1 +n2 −1, k1 +k2 −1, n3 , k3 )+(k2 n3 −k3 n2 )c(n2 +n3 −1, k2 +k3 −1, n1 , k1 )+ +(k3 n1 − k1 n3 )c(n3 + n1 − 1, k3 + k1 − 1, n2 , k2 ) ¤
Remark 1. We notice that the sum of the first and third (resp. second and fourth) arguments in the three 2-cocycle values c(n2 + n3 − 1, k2 + k3 − 1; n1 , k1 ), c(n1 + n2 − 1, k1 + k2 − 1; n3 , k3 ) and c(n3 + n1 − 1, k3 + k1 − 1; n2 , k2 ) appearing in (3.7) is equal to n1 + n2 + n3 − 1 (resp. k1 + k2 + k3 − 1). Proposition 1. Let S, M ∈ {−1, 0, 1, ...} be given and let n1 , k1 , n2 , k2 , n3 , k3 be nonnegative integers satisfying (3.8)
n1 + n2 + n3 = S + 1 ; k1 + k2 + k3 = M + 1
For i ∈ {1, 2, 3} let (3.9)
ψ(ni , ki ) = c(S − ni , M − ki ; ni , ki )
with the skew-symmetry condition (derived from (3.1), (3.3) and (2.3)) (3.10)
ψ(ni , ki ) = −ψ(S − ni , M − ki )
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LUIGI ACCARDI AND ANDREAS BOUKAS
Then, all analytic (in the sense of definition 1) solutions ψ of ( the ψ–form of (3.7)) (3.11)
(k2 n3 − k3 n2 ) ψ(n1 , k1 ) + (k3 n1 − k1 n3 ) ψ(n2 , k2 ) + (k1 n2 − k2 n1 ) ψ(n3 , k3 ) = 0
i.e all ψ satisfying (3.11) for all n1 , k1 , n2 , k2 , n3 , k3 satisfying (3.8), are of the form (3.12)
ψ(ni , ki ) = z (M ni − S ki )
where z = zS,M ∈ C. Note: The dependence of z on S and M is justified by the fact that, in accordance with Remark 1 above, all ψ’s appearing in (3.11) must have the same z. Proof. In view of (3.9) and Definition 1, let ψ(ni , ki ) =
X
ar,ρ nri kiρ
r,ρ≥0
be an analytic solution of (3.11). Then (3.11) becomes a0,0 (k2 n3 −k3 n2 +k3 n1 −k1 n3 +k1 n2 −k2 n1 )+a1,0 ·0+a0,1 ·0+
X
ar,ρ fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 )
r+ρ>1
where fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 ) = (k2 n3 − k3 n2 ) nr1 k1ρ + (k3 n1 − k1 n3 ) nr2 k2ρ + (k1 n2 − k2 n1 ) nr3 k3ρ We notice that, if M 6= −1 then fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 ) 6≡ 0 for all n1 , k1 , n2 , k2 , n3 , k3 satisfying (3.8). Since k2 n3 − k3 n2 + k3 n1 − k1 n3 + k1 n2 − k2 n1 6≡ 0 either, it follows that a0,1 and a1,0 are arbitrary complex numbers and a0,0 = ar,ρ = 0 for r + ρ > 1. If M = −1 then k1 + k2 + k3 = 0 implies that k1 = k2 = k3 = 0 and so, by (3.6), ψ(n1 , k1 ) = 0 which is of the form (3.12) with z = 0. Thus (3.13)
ψ(ni , ki ) = a1,0 ni + a0,1 ki
By (3.10) ψ(n1 , k1 ) = −ψ(S − n1 , M − k1 ) which combined with (3.13) implies that (3.14)
a1,0 S + a0,1 M = 0
Case (i). S 6= 0 and M 6= 0: Then (3.14) implies that a1,0 = −a0,1 z ∈ C, we find that a1,0 = z M and (3.13) becomes
M S
and letting a0,1 = −z S,
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
7
ψ(ni , ki ) = z (M ni − S ki ) Case (ii). S = 0 and M = 0: Then n1 + n2 + n3 = 1 and k1 + k2 + k3 = 1 and so one of the ni and one of the ki are equal to 1 and the rest are equal to 0. If (n1 , n2 , n3 ) = (1, 0, 0) and (k1 , k2 , k3 ) = (1, 0, 0) then ψ(n1 , k1 ) = c(n2 + n3 − 1, k2 + k3 − 1; n1 , k1 ) = c(−1, −1; 1, 1) = 0 ( by (3.6)) ψ(n2 , k2 ) = c(n3 + n1 − 1, k3 + k1 − 1; n2 , k2 ) = c(0, 0; 0, 0) = 0 ( by (3.5)) ψ(n3 , k3 ) = c(n1 + n2 − 1, k1 + k2 − 1; n3 , k2 ) = c(0, 0; 0, 0) = 0 ( by (3.5)) Thus ψ(ni , ki ) = 0 = z (0 ni − 0 ki ) in agreement with (3.12). The cases (n1 , n2 , n3 ) = (0, 1, 0), (k1 , k2 , k3 ) = (0, 1, 0) and (n1 , n2 , n3 ) = (0, 0, 1), (k1 , k2 , k3 ) = (0, 0, 1) are similar. If (n1 , n2 , n3 ) = (1, 0, 0) and (k1 , k2 , k3 ) = (0, 1, 0) then ψ(n1 , k1 ) = c(n2 + n3 − 1, k2 + k3 − 1; n1 , k1 ) = c(−1, 0; 1, 0) = 0 ( by (3.6)) ψ(n2 , k2 ) = c(n3 + n1 − 1, k3 + k1 − 1; n2 , k2 ) = c(0, −1; 0, 1) = 0 ( by (3.6)) ψ(n3 , k3 ) = c(n1 + n2 − 1, k1 + k2 − 1; n3 , k2 ) = c(0, 0; 0, 0) = 0 ( by (3.5)) Thus ψ(ni , ki ) = 0 = z (0 ni − 0 ki ) in agreement with (3.12). All other cases ni0 = 1, ni 6= 0, i 6= i0 , and kj0 = 1, nj 6= 0, j 6= j0 , where i0 6= j0 , are similar. Case (iii). S = 0 and M 6= 0: Then, by (3.14), a0,1 = 0 and by (3.13) ψ(ni , ki ) = a1,0 ni = z (M ni − 0 ki ) 1,0 in agreement with (3.12) for z = aM . Case (iv). S 6= 0 and M = 0: Then, by (3.14), a1,0 = 0 and by (3.13) ψ(ni , ki ) = a0,1 ki = z (0 ni − S ki ) in agreement with (3.12) for z = − aS0,1 . Corollary 1. Let n, k, N, K be nonnegative integers. Then c(n, k; N, K) = zn+N,k+K · (k N − K n) where zn+N,k+K ∈ R.
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LUIGI ACCARDI AND ANDREAS BOUKAS
Proof. By proposition 1 c(n1 + n2 − 1, k1 + k2 − 1; n3 , k2 ) = z ((k1 + k2 − 1) n3 − (n1 + n2 − 1) k2 ) which letting n1 + n2 − 1 = n, k1 + k2 − 1 = k, n3 = N and k3 = K implies that c(n, k; N, K) = z (k N − n K) where, by proposition 1, z = zn+N,k+K ∈ C. By (3.4), z = z¯ and so zn+N,k+K ∈ R. ¤ Theorem 1. All analytic central extensions (3.2) of the 1–mode RHPWN ∗–Lie algebra are trivial. Proof. Define a complex valued function f on the 1–mode RHPWN generators by f (Bkn ) = zn+1,k+1 where zn+1,k+1 is as in Corollary 1, and extend by linearity the definition of f to the whole RHPWN algebra. Then ¡ ¢ n+N −1 N f ([Bkn , BK ]RHP W N ) = f (k N − K n) Bk+K−1 ¡ n+N −1 ¢ = (k N − K n) f Bk+K−1 = (k N − K n) zn+N,k+K = c(n, k; N, K) N = φ(Bkn , BK ) Thus the central extension is trivial. ¤ 4. Analytic central extensions of the 1–mode w∞ Lie algebra Denote by w g ∞ an analytic central extension of the 1–mode w∞ Lie algebra such that on the w∞ generators ˆ n, B ˆ N ]wg ˆn ˆN [B k K ∞ = [Bk , BK ]w∞ + c(n, k; N, K) E ˆ n+N −2 + c(n, k; N, K) E = (k (N − 1) − K (n − 1)) B k+K
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: ˆ n, B ˆN ) ∈ C c(n, k; N, K) = φ(B k K 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) = −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
ˆ n equation (2.4) is equivalent to Lemma 2. On the 1–mode w∞ generators B k (4.4)
(k2 (n3 − 1) − k3 (n2 − 1)) c(n2 + n3 − 2, k2 + k3 ; n1 , k1 ) +(k3 (n1 − 1) − k1 (n3 − 1)) c(n3 + n1 − 2, k3 + k1 ; n2 , k2 ) +(k1 (n2 − 1) − k2 (n1 − 1)) c(n1 + n2 − 2, k1 + k2 ; n3 , k3 ) = 0
Proof. The proof is similar to that of lemma 1
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Proposition 2. Let S ≥ 0 and M ∈ Z be given and let n1 , k1 , n2 , k2 , n3 , k3 ∈ Z satisfy n1 , n2 , n3 ≥ 2 and (4.5)
n1 + n2 + n3 = S + 2 ; k1 + k2 + k3 = M
For i ∈ {1, 2, 3} let ψ(ni , ki ) = c(S − ni , M − ki ; ni , ki ) where by (4.1) (4.6)
ψ(ni , ki ) = −ψ(S − ni , M − ki )
Then, all analytic (in the sense of definition 1) solutions ψ of ( the ψ–form of (4.4)) (4.7)
(k2 (n3 − 1) − k3 (n2 − 1)) ψ(n1 , k1 ) + (k3 (n1 − 1) − k1 (n3 − 1)) ψ(n2 , k2 ) +(k1 (n2 − 1) − k2 (n1 − 1)) ψ(n3 , k3 ) = 0
i.e all ψ satisfying (4.7) for all n1 , k1 , n2 , k2 , n3 , k3 ∈ Z satisfying n1 , n2 , n3 ≥ 2 and (4.5), are of the form (4.8)
ψ(ni , ki ) = z (M (ni − 1) − (S − 2) ki )
where z = zS,M ∈ C. Proof. Let ψ(ni , ki ) =
X r,ρ≥0
Then (4.7) becomes
ar,ρ (ni − 1)r kiρ
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LUIGI ACCARDI AND ANDREAS BOUKAS
a0,0 (k2 (n3 − 1) − k3 (n2 − 1) + k3 (n1 − 1) − k1 (n3 − 1) + k1 (n2 − 1) − k2 (n1 − 1)) P +a1,0 · 0 + a0,1 · 0 + r+ρ>1 ar,ρ fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 ) = 0 where fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 ) = (k2 (n3 − 1) − k3 (n2 − 1)) (n1 − 1)r k1ρ + (k3 (n1 − 1) − k1 (n3 − 1)) (n2 − 1)r k2ρ +(k1 (n2 − 1) − k2 (n1 − 1)) (n3 − 1)r k3ρ We notice that fr,ρ (n1 , n2 , n3 , k1 , k2 , k3 ) 6≡ 0 for all n1 , k1 , n2 , k2 , n3 , k3 satisfying (4.5). Since the same is true for the coefficient of a0,0 , it follows that a0,1 and a1,0 are arbitrary complex numbers and a0,0 = ar,ρ = 0 for r + ρ > 1. Thus (4.9)
ψ(ni , ki ) = a1,0 (ni − 1) + a0,1 ki
which combined with (4.6) implies that (4.10)
a1,0 (S − 2) + a0,1 M = 0
Case (i). S 6= 2 and M 6= 0: Then (4.10) implies that a1,0 = −a0,1 −z (S − 2), z ∈ C, we find that a1,0 = z M and (4.9) becomes
M S−2
and letting a0,1 =
ψ(ni , ki ) = z (M (ni − 1) − (S − 2) ki ) Case (ii). S = 2 and M = 0: Then n1 + n2 + n3 = 4 and k1 + k2 + k3 = 0 and so, since n1 , n2 , n3 ≥ 2, without loss of generality, either (n1 , n2 , n3 ) = (2, 2, 0) or (n1 , n2 , n3 ) = (4, 0, 0). Therefore, in (4.7) we are reduced to ψ(2, k1 ), ψ(2, k2 ) and ψ(0, −(k1 + k2 )) or to ψ(4, k1 ), ψ(0, k2 ) and ψ(0, −(k1 + k2 )) i.e. to c(0, −k1 ; 2, k1 ), c(0, −k2 ; 2, k2 ) and c(2, k1 + k2 ; 0, −(k1 + k2 )) or to c(−2, −k1 ; 4, k1 ), c(2, −k2 ; 0, k2 ) and c(2, k1 + k2 ; 0, −(k1 + k2 )) which, by (4.3), are all equal to zero . Thus, in this case, ψ(ni , ki ) = 0 = z (0 (ni − 1) − 0 ki ) in agreement with (4.8). Case (iii). S 6= 2 and M = 0: Then a1,0 (S − 2) + a0,1 · 0 = 0 which implies that a1,0 = 0 i.e.
where z =
a0,1 2−S
ψ(ni , ki ) = a0,1 ki = z (0 (ni − 1) − (S − 2) ki ) in agreement with (4.8).
Case (iv). S = 2 and M 6= 0: Then a1,0 · 0 + a0,1 M = 0 which implies that a0,1 = 0 i.e. ψ(ni , ki ) = a1,0 (ni − 1) = z (M (ni − 1) − 0 ki )
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
where z =
a1,0 M
in agreement with (4.8).
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Corollary 2. Let n, k, N, K be integers, with n, N ≥ 2. Then c(n, k; N, K) = zn+N,k+K · (k (N − 1) − K (n − 1)) where zn+N,k+K ∈ R. Proof. By proposition 2 c(n1 + n2 − 2, k1 + k2 ; n3 , k3 ) = z ((k1 + k2 ) (n3 − 1) − (n1 + n2 − 3) k3 ) which letting n1 + n2 − 2 = n, k1 + k2 = k, n3 = N and k3 = K implies that c(n, k; N, K) = z (k (N − 1) − (n − 1) K) where, by proposition 2, z = zn+N,k+K ∈ C. By (4.2) z (k (N − 1) − (n − 1) K) = −z ((−k) (N − 1) − (n − 1) (−K)) = z¯ (k (N − 1) − (n − 1) K) i.e. z = z¯ which implies that zn+N,k+K ∈ R. ¤ Theorem 2. All analytic central extensions of the 1–mode w∞ ∗–Lie algebra are trivial. Proof. Define a complex valued function f on the 1–mode w∞ generators by ˆ n ) = zn+2,k f (B k where zn+2,k is as in Corollary 2. Linearly extend the definition of f to all of w∞ . Then ³ ´ N ˆkn , B ˆK ˆ n+N −2 f ([B ]w∞ ) = f (k (N − 1) − K (n − 1)) B k+K ´ ³ n+N −2 ˆ = (k (N − 1) − K (n − 1)) f Bk+K = (k (N − 1) − K (n − 1)) zn+N,k+K = c(n, k; N, K) ˆ n, B ˆN ) = φ(B k
K
Thus the central extension is trivial. ¤
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LUIGI ACCARDI AND ANDREAS BOUKAS
5. Appendix: Analytic central extensions of the Witt Lie algebra It is well known (cf. [10]) that the (1–mode) Witt algebra W with generators {Lm ; m ∈ Z} satisfying the commutation relations (5.1)
[Lm , Ln ]W = (m − n) Lm+n
with adjointness condition (Lm )∗ = L−m can be centrally extended to the Virasoro algebra V through (5.2)
[Lm , Ln ]V = (m − n) Lm+n + δm+n,0 m (m2 − 1) E
The Witt algebra with commutation relations (5.1) is a sub-algebra of the w∞ Lie algebra with commutation relations (1.3). In Proposition 4 below we describe how the technique used in the proof of proposition 1 can be used to derive (5.2). Our approach differs from the traditional one, where recursions are constructed so as to yield (5.2), in that we obtain all non-trivial analytic central extensions of the Witt algebra which we then show to be equivalent to (5.2). f be an analytic central extension of the Witt ∗–Lie algebra such that on the W So let W generators [Lk , LK ]W f = (k − K) Lk+K + c(2, k; 2, K) E where k, K ∈ Z c(2, k; 2, K) = φ(Lk , LK ) ∈ C where φ is a cocycle as in (2.6). 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) = −c(2, −k; 2, −K)
Proposition 3. The Virasoro central extension (5.2) of the Witt algebra (5.1) is non-trivial.
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
13
Proof. Suppose that (5.2) is a trivial extension of (5.1). Then there exists a linear complexvalued function f defined on the Witt algebra such that f ([Lm , Ln ]W ) = δm+n,0 m (m2 − 1)
(5.5)
By (5.1) and the linearity of f , equation (5.5) yields (m − n) f (Lm+n ) = δm+n,0 m (m2 − 1) from which for n = 0 we obtain m f (Lm ) = δm,0 m (m2 − 1) which implies that ½
0 if m 6= 0 z if m = 0 where z ∈ C is arbitrary. But then, for any m 6= 0, f (Lm ) =
µ
(5.6)
1 z = f (L0 ) = f [Lm , L−m ]W 2m 1 m2 − 1 = m (m2 − 1) = 2m 2
¶ =
1 f ([Lm , L−m ]W ) 2m
and also µ
¶ 1 1 z = f (L0 ) = f (5.7) [L2 m , L−2 m ]W = f ([L2 m , L−2 m ]W ) 4m 4m 1 4 m2 − 1 = 2 m (4 m2 − 1) = 4m 2 By (5.6) and (5.7) it follows that f (L0 ) cannot be consistently defined. Thus such an f cannot exist, therefore the central extension (5.2) of (5.1) is non-trivial. ¤ ˆ 2 in the notation of w∞ ) equation (2.4) Lemma 3. On the Witt algebra generators Lk (= B k is equivalent to (5.8) (k2 −k3 ) c(2, k2 +k3 ; 2, k1 )+(k3 −k1 ) c(2, k3 +k1 ; 2, k2 )+(k1 −k2 ) c(2, k1 +k2 ; 2, k3 ) = 0 Proof. The proof is identical to that of lemma 1 Proposition 4. (i) The Witt algebra (5.1) analytic central extensions are given by (5.9)
2 [Lm , Ln ]W f = (m − n) Lm+n + δm+n,0 m (β m + α) E
¤
14
LUIGI ACCARDI AND ANDREAS BOUKAS
where α, β ∈ R. (ii) The analytic central extensions (5.9) are trivial if and only if β = 0. (iii) All analytic central extensions of the form (5.9) are equivalent to the Virasoro central 1 extension (5.2) which corresponds to the choice α = −β = − 12 . Proof. (i) Letting θ(ki ) = c(2, −ki ; 2, ki ) (5.8) becomes (k2 − k3 ) θ(k1 ) + (k3 − k1 ) θ(k2 ) + (k1 − k2 ) θ(k3 ) = 0 which, assuming that X θ(ki ) = aρ kiρ ρ≥0
yields that for all integers k1 , k2 , k3 such that k1 + k2 + k3 = 0 X aρ ((k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ ) = 0 ρ≥0
Clearly for ρ = 0 and ρ = 1 (k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ ≡ 0 For ρ = 3
= = = =
(k2 − k3 ) k13 + (k3 − k1 ) k23 + (k1 − k2 ) k33 k2 k1 (k12 − k22 ) + k3 k2 (k22 − k32 ) + k1 k3 (k32 − k12 ) k2 k1 (k1 + k2 ) (k1 − k2 ) + k3 k2 (k2 + k3 ) (k2 − k3 ) + k1 k3 (k3 + k1 ) (k3 − k1 ) −k2 k1 k3 (k1 − k2 ) + k3 k2 k1 (k2 − k3 ) + k1 k3 k2 (k3 − k1 ) −k2 k1 k3 (k1 − k2 + k2 − k3 + k3 − k1 ) ≡ 0
Thus a0 , a1 , a3 are arbitrary. For ρ 6= 0, 1, 3 (k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ 6≡ 0 Thus all other aρ ’s are equal to zero and so θ(ki ) = a0 + a1 ki + a3 ki3 By (5.3) θ(ki ) = c(2, −ki ; 2, ki ) = −c(2, ki ; 2, −ki ) = −θ(−ki ) which implies that a0 = 0. Similarly, by (5.4)
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
15
θ(ki ) = c(2, −ki ; 2, ki ) = −c(2, ki ; 2, −ki ) = −θ(−ki ) which implies that for all integers ki a1 ki + a3 ki3 = a ¯ 1 ki + a ¯3 ki3 and so a1 = a ¯1 and a3 = a ¯3 i.e. a1 = β and a3 = α where α, β ∈ R. We have therefore proved (5.9) for n + m = 0. To prove it for n + m 6= 0 we let n1 = n2 = n3 = 2 in (5.8) and, assuming that k1 + k2 + k3 = L 6= 0, we obtain (k2 − k3 ) c(2, L − k1 ; 2, k1 ) + (k3 − k1 ) c(2, L − k2 ; 2, k2 ) + (k1 − k2 ) c(2, L − k3 ; 2, k3 ) = 0 which letting θ(ki ) = c(2, L − ki ; 2, ki ) becomes (k2 − k3 ) θ(k1 ) + (k3 − k1 ) θ(k2 ) + (k1 − k2 ) θ(k3 ) = 0 which assuming that X θ(ki ) = aρ kiρ ρ≥0
yields that for all integers k1 , k2 , k3 such that k1 + k2 + k3 = L X aρ ((k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ ) = 0 ρ≥0
As before, for ρ = 0 and ρ = 1 (k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ ≡ 0 and so a0 , a1 are arbitrary. For ρ ≥ 2 (k2 − k3 ) k1ρ + (k3 − k1 ) k2ρ + (k1 − k2 ) k3ρ 6≡ 0 thus aρ = 0 for all ρ ≥ 2 and so θ(ki ) = a0 + a1 ki By (5.3) θ(ki ) = c(2, L − ki ; 2, ki ) = −c(2, ki ; 2, L − ki ) = −θ(L − ki ) implies that a0 + a1 ki = −a0 − a1 (L − ki ), i.e. 2 a0 + a1 L = 0 and so, since L is arbitrary, a0 = a1 = 0. Thus θ(ki ) ≡ 0. (ii) Let φ(Ln , Lm ) = δn+m,0 (α m + β m3 ). If β = 0 then the linear extension of the function
16
LUIGI ACCARDI AND ANDREAS BOUKAS
α 2 to all of the Witt algebra satisfies f ([Ln , Lm ]W ) = φ(Ln , Lm ) and so the extension is trivial. Conversely, if β 6= 0 and f is a linear function satisfying f ([Ln , Lm ]W ) = φ(Ln , Lm ) then f (Ln ) = −δn,0
(n − m) f (Ln+m ) = δn+m,0 (α m + β m3 ) which for n + m = 0, m 6= 0 and n 6= m implies that α + β m2 −2 which means that f (L0 ) cannot be consistently defined. Therefore such an f does not exist and so the extension is not trivial. f (L0 ) =
1 (iii) As pointed out already, the factor 12 in (5.2) is for traditional reasons only and can therefore be replaced by an arbitrary real number β 6= 0. We may thus write (5.2) and (5.9) as
[Lm , Ln ]V = (m − n) Lm+n + δm+n,0 (m3 − m) β E and 3 [Lm , Ln ]W f = (m − n) Lm+n + δm+n,0 (m +
α m) β E β
respectively. Letting φ(Ln , Lm ) = δn+m,0 (m3 +
α m) β
and ψ(Ln , Lm ) = δn+m,0 (m3 − m) we see that ω(Ln , Lm ) = (φ − ψ)(Ln , Lm ) = δn+m,0 (
α + 1) m β
is a trivial cocycle since ω(Ln , Lm ) = f ([Ln , Lm ]W ) where f (Lk ) = −δk,0 Thus φ and ψ are equivalent cocycles.
α β
+1 2 ¤
ANALYTIC CE OF INFINITE DIMENSIONAL WN ∗–LIE ALGEBRAS
17
Remark 2. From the purely algebraic point of view, i.e. forgetting about its white noise origin, the RHPWN ∗–Lie algebra (1.2) 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 Bk1 where k ∈ Z (in analogy to ˆ 2 ). Taking n1 = n2 = n3 = 1 and M = −1 in (3.11) we obtain the Witt algebra generators B k (5.10)
(k2 − k3 ) ψ(1, k1 ) + (k3 − k1 ) ψ(1, k2 ) + (k1 − k2 ) ψ(1, k3 ) = 0 P where ψ(1, ki ) = −ψ(1, −1 − ki ) and so, letting θ(ki ) = ψ(1, ki ) = ρ≥0 aρ kiρ , we have the skew-symmetry condition θ(ki ) = −θ(−1 − ki ). Substituting in (5.10) and expanding we find that X aρ ((k2 − k3 )k1ρ + (k3 − k1 )k2ρ + (k1 − k2 )k3ρ ) a0 · 0 + a1 · 0 + a3 · 0 + ρ6=0,1,3
where in the summation on the right hand side, (k2 − k3 )k1ρ + (k3 − k1 )k2ρ + (k1 − k2 )k3ρ 6≡ 0. Thus for all ρ 6= 0, 1, 3 we have that aρ ≡ 0 and so θ(ki ) = a0 + a1 ki + a3 ki3 . By the above skew-symmetry condition it follows that a3 = 0 and a1 = 2 a0 . Therefore, letting a1 = λ, we have that θ(ki ) = λ2 + λ ki which is of the form (3.12) with z = − λ2 . 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, (2006) 353360. [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, (2006) 315–342. [3] : Renormalized Higher Powers of White Noise and the Virasoro–Zamolodchikov–w∞ Algebra, Reports on Mathematical Physics, Volume 61, number 1, 2008, pages 1-11, http://arxiv.org/hepth/0610302. [4] : Fock representation of the renormalized higher powers of white noise and the Virasoro– Zamolodchikov–w∞ ∗–Lie algebra, to appear in Journal of Physics A: Mathematical and Theoretical (2007), arXiv:0706.3397v2 [math-ph]. [5] : Lie algebras associated with the renormalized higher powers of white noise, Communications on Stochastic Analysis Vol. 1, No. 1, 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, 129–147, (2006) [7] Bakas, I.: The structure of the Winf algebra, Commun. Math. Phys. 134 (1990) 487-508 [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)
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[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 (1998) 2569–2577 ` di Roma Tor Vergata, via Columbia 2, 00133 Roma, Centro Vito Volterra, Universita Italy E-mail address:
[email protected] URL: http://volterra.mat.uniroma2.it Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi, Athens 15342, Greece E-mail address:
[email protected]
