On Solvable Lie Algebras of White Noise Operators

: We characterize the dimension of Lie algebras of white noise operators containing the quantum white noise derivatives of the conservation operator. We establish isomorphisms to ﬁliform Lie algebras, Engel-type algebras, and solvable Lie algebras with Heisenberg nilradical and Abelian nilradical. A new class of solvable Lie algebras is proposed, those having an Engel-type algebra as nilradical. This arises in white noise analysis as a 2 n + 3-dimensional Lie algebra containing the identity operator, annihilation operators, creation operators (Heisenberg algebra), number operator, and Gross Laplacian.


Introduction
Lie algebras typically arise as vector spaces of linear transformations endowed with a binary operation, which is in general, neither commutative nor associative.A Lie algebra over a field K is a vector space L over K, with an operation L × L → L, denoted (x, y) → [x, y], which is called the bracket or commutator of x and y, that satisfies the following conditions: (L1) The bracket operation is bilinear.(L2) [x, x] = 0 for all x ∈ L. (L3) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ L (Jacobi identity).A Lie algebra L is abelian if [x, y] = 0 for all x, y ∈ L. We say that two Lie algebras L and L over K are isomorphic if there exists a vector space isomorphism φ : L → L satisfying φ([x, y]) = [φ(x), φ(y)] for all x, y ∈ L. A subspace I of a Lie algebra L is called an ideal of L if x ∈ L, y ∈ I together imply [x, y] ∈ I.The derived series is defined to be a sequence of ideals of L given by L (0) = L, L (1) = [L, L], L (2) = [L (1) , L (1) ], . . ., L (i) = [L (i−1) , L (i− 1) ].
A Lie algebra L is solvable if L (n) = 0 for some n.Moreover, the lower central series is defined as a sequence of ideals of L given by A Lie algebra L is nilpotent if L n = 0 for some n.The maximal nilpotent ideal of a Lie algebra is called its nilradical.For more details on Lie algebras see [1].Sophus Lie introduced Lie groups to study symmetries of solutions of differential equations.Their corresponding Lie algebras are vector fields generating one-parameter subgroups.Since Lie algebras are linear spaces they are easier to study than the Lie groups themselves.They have several applications in mathematics and related fields such as mathematical physics [2], stochastic calculus [3], and robotics [4], to name a few.For instance, they arise as vector spaces of linear operators which commute with a given operator, say, the Hamiltonian of a physical system [5].
One particular area of mathematics where Lie algebras are used is in white noise analysis, which was initiated by Hida [6] in 1975 to realize the time derivative of Brownian motion.It has been applied to other areas of mathematics such as mathematical physics and finance, see e.g., [7][8][9].White noise analysis is an infinite dimensional analog of Schwartz' distribution theory, where the roles of the Lebesgue measure on R d and the Gelfand triple are played by the Gaussian measure µ on E * and respectively.We have the following expressions of the finite dimensional Laplacian ∆ on R d : when ∆ acts on S(R d ).The Gross Laplacian and number operator are infinite dimensional analogs of the Laplacian ∆ expressed as follows: However, unlike the finite dimensional case, ∆ G and N are completely different from each other [10].Moreover, ∆ G and N are characterized by their rotation-invariance [11].Oneparameter transformation groups acting on the space of test white noise functionals (E) and the Lie algebra spanned by their infinitesimal generators are discussed in [12].They obtained a five-dimensional complex Lie algebra generated by the identity operator, number operator, Gross Laplacian, and the infinitesimal generators of differentiation and multiplication operators.A two-parameter transformation group has been constructed in [13] with corresponding Lie algebra C∆ G + CN.Later on, multi-parameter transformation groups [14] and Laplacians of diagonal type [10] were also studied.The generalized Gross Laplacian was introduced in [15].Ji and Sinha [16] studied a six-dimensional non-solvable Lie * -algebra containing the conservation operator and generalized Gross Laplacian as well as the quantum stochastic processes they induce.The classification of solvable Lie algebras of finite dimension still remains unsolved.A realistic partial classification problem is to classify all solvable Lie algebras with a given nilradical [5].The goal of this paper is to investigate Lie algebras of white noise operators and show that they are isomorphic to some known abstract Lie algebra.In the course of this investigation, we describe a Lie algebra of white noise operators that belongs to the class of solvable Lie algebras with Engel-type nilradical.This paper is organized as follows: In Section 2, we assemble standard notations used in white noise analysis.Section 3 is devoted to a review of classification of solvable Lie algebras.In Section 4, we prove isomorphisms to Lie algebras with Heisenberg nilradical and Engel-type nilradical.A solvable Lie algebra with Engel-type nilradical arises from the extension of the Heisenberg Lie algebra by the Gross Laplacian and number operator.In Section 5, we characterize the dimension of Lie algebras of white noise operators containing the quantum white noise derivatives of the conservation operator.In Section 6, we establish isomorphisms to filiform Lie algebras.Section 7 is devoted to the study of white noise Lie algebras with Abelian nilradicals.In particular, we show that the Lie algebra generated by the creation operator and the conservation operator corresponding to the Fourier-Mehler transform (fractional Fourier transform) is parametrized by the nth roots of unity.In Section 8, we show the linear independence of generalized Gross Laplacians (quadratic annihilation operators) corresponding to the nth powers of the unilateral shift.

Preliminaries
In this section, we introduce standard definitions and notations used in white noise analysis, for more details see [7,8,11].To outline this section, we start with a real Gelfand triple where S(R) is the Schwartz space of rapidly decreasing functions, S (R) is the space of tempered distributions, and we identify H with its dual space H * by Riesz Representation Theorem.We denote the canonical bilinear form on E * × E and the inner product of H by the same symbol •, • since they are compatible.Then, we introduce a measure on E * via Bochner-Minlos theorem.Finally, spaces of test white noise functionals and generalized white functionals are constructed in Section 2.1.2.Moreover, some properties of the exponential vectors are enumerated.

Standard Countably Hilbert Space Construction
We recall the following definitions; for more details see [8].We say that a topological vector space V together with a family {| • | n } of inner product norms is a countably Hilbert space or CH-space if V is complete with respect to its topology.Suppose that V is a CHspace together with an increasing sequence {| • | n } of norms.Denote by V n the completion V with respect to the norm | • | n .A CH-space V is said to be a nuclear space if for any n, there exists m ≥ n such that the inclusion map from V m into V n is a Hilbert-Schmidt operator.There are several methods of constructing CH-spaces (see, e.g., [7,11]) but we will focus on the construction given a Hilbert space and an operator satisfying certain conditions.Since our results deal with the Schwartz space S(R) and the space of test white noise functionals (E), the constructions in Sections 2.1.1 and 2.1.2are sufficient for our purposes.

Reconstruction of Schwartz Space
As an example of a standard countably Hilbert space, we will reconstruct the Schwartz space S(R) from the real Hilbert space H = L 2 (R) and the operator Recall that a function ξ on R is a rapidly decreasing function if it is smooth and for all nonnegative integers n and k, |x n ξ (k) (x)| → 0 as |x| → ∞.Then S(R) is defined as the space of rapidly decreasing functions on R. For any n, k ≥ 0, define a norm This family {| • | n,k ; n, k ≥ 0} of norms generates a topology on S(R).Hence, Schwartz space S(R) is a topological vector space.The Hermite polynomial of degree n is defined as is the corresponding Hermite function.Then the Hermite functions form an orthonormal basis for L 2 (R).In fact, they are eigenfunctions of A since ∀n ≥ 0, Ae n = (2n + 2)e n .Since A is injective, its inverse A −1 : R(A) → L 2 (R) exists with the following eigenvalues A −1 = (2n + 2) −1 e n .Since S(R) is contained in the range of A and is dense in L 2 (R), we can extend A to L 2 (R).We have that For any p > 1/2, A −p is a Hilbert-Schmidt operator of L 2 (R).In fact, its Hilbert-Schmidt norm is given by For each p ≥ 0, define The families {| • | n,k ; n, k ≥ 0} and {| • | p ; p ≥ 0} are equivalent, i.e, they generate the same topology on S(R).

3.
S(R) is a nuclear space.
We have reconstructed S(R) as the projective limit of {S p (R); p ≥ 0}, i.e., We also have the following continuous inclusions: It is known that the norm of the dual space S p (R) of S p (R) is given by and that S p (R) is the completion of L 2 (R) with respect to the norm | • | −p .Moreover, S (R) is the dual space of S(R) and S p (R) = ind lim p→∞ S p (R).

Spaces of Test and Generalized White Noise Functionals
Given a locally convex space X over R, denote by X C its complexification.Then, we extend the canonical bilinear form X * × X, which is an R-bilinear form, to a C-bilinear form on X * C × X C , denoted by the same symbol [11].As before, we start with a real Gelfand triple Applying Bochner-Minlos theorem [8], a probability measure µ on E * exists such that The measure µ is called the standard Gaussian measure on E * and the probability space (E * , µ) is called the white noise space.The following construction of test and generalized white noise functionals follows from [8], pp.18-20.Let us denote by (L 2 ) the complex Hilbert space L 2 (E * , µ; C).As before, we let The (L 2 )-norm ||φ|| 0 of φ is given by , where | • | 0 denotes the L 2 C (R n )-norm for any n.We now use the second quantization operator of A, denoted by Γ(A), as follows.For φ ∈ (L 2 ) satisfying the condition that This operator is densely defined on (L 2 ) and has the following properties similar to A in Section 2.1.1: 1.
(L 2 ) has an orthonormal basis consisting of eigenfunctions of Γ(A).

3.
For p > 1, Γ(A) −p is a Hilbert-Schmidt operator of (L 2 ).Following the same procedure in Section 2.1.1 we construct a space of test white noise functionals.For p ≥ 0, define Then (E p ) is a Hilbert space with norm || • || p .Moreover, we define which we call the space of test white noise functionals.Its dual space (E) * is the space of generalized white noise functionals.The bilinear pairing of (E) * and (E) is denoted by •, • .It is related to the inner product of (L 2 ) by

Exponential Vectors
We often have to deal with operators which are determined uniquely by their action on the exponential vectors.Here, we will enumerate some properties of the exponential vectors.Let h be a real or complex Hilbert space with the norm | the Hilbert space Γ(h) is called the (Boson) Fock space or the symmetric Hilbert space over h [11].For ξ ∈ h we put Then and, in particular, φ ξ ∈ Γ(h).We call φ ξ in (5) an exponential vector.

White Noise Operators
The pair of annihilation and creation operators {a t , a * t : t ∈ T} on a time parameter space T is called a quantum white noise (field) on T [17].For y ∈ E * , we define These are called the annihilation and creation operators associated with y, respectively.For ξ ∈ E, it is known (see e.g., [10,11]) that the action of the annihilation operator a(y) on the exponential vectors is given by a(y)φ ξ = y, ξ φ ξ .
If ζ ∈ E, then a(ζ) can be extended to (E) * so that it belongs to L((E) * , (E) * ), denoted by the same symbol.On the other hand, the restriction of a * (ζ) to (E) belongs to L((E), (E)) [18].Thus, given any white noise operator are well-defined and belongs to L((E), (E) * ).We define

Definition 1 ([18]
). D + ζ Ξ and D − ζ Ξ are called the creation derivative and annihilation derivative of Ξ, respectively.They are collectively called the quantum white noise derivatives (qwn-derivatives) of Ξ.The quantum white noise derivatives of the generalized Gross Laplacian and conservation operator are given in [17,19].By the kernel theorem, an operator S ∈ L(E, E * ), The generalized Gross Laplacian associated with S is the integral kernel operator given by It is known that ∆ G (S) ∈ L((E), (E)).We recover the usual Gross Laplacian by taking S = Id, i.e., ∆ G = ∆ G (Id).The conservation operator associated with S is the integral kernel operator Generally, Λ(S) ∈ L((E), (E) * ).It is known that Λ(S) ∈ L((E), (E)) if and only if S is a continuous linear operator on E C [11].We recover the number operator by taking S = Id, i.e., N = Λ(Id).
Then h is a five-dimensional non-nilpotent solvable complex Lie algebra.

Classification of Lie Algebras
We recall the classification of Lie algebras relevant to our study.This helps us classify white noise Lie algebras and propose a new class of solvable Lie algebras, those having an Engel-type algebra as nilradical.

Heisenberg Lie Algebras and Their extensions
The Heisenberg algebra H(n) is defined by its standard basis {p 1 , . . ., p n , q 1 , . . ., q n , h} with commutation relations It is of primordial importance in quantum mechanics.Extensions of H(n) by further operators s 1 , . . ., s f is a question of the algebra of quantum mechanical observables.These extensions are denoted by L(n, f ).The construction of all finite dimensional indecomposable solvable Lie algebras L(n, f ) containing H(n) as nilradical was performed in [20].They also obtained the Casimir invariants of the algebras L(n, f ).Solvable extensions of the Heisenberg algebra H(n) by the number operator are denoted as M (3) (2.21a) and F 2 (0, 1, 1) of Table A1 in [20], and as s 4,6 and s 6,162 with a = 1 in [5].The connected Lie groups with Lie algebra s 4,6 can be found in [21] Cor.4.10.
An element of the Lie algebra of the symplectic group Sp(1, R) induces a derivation of the Heisenberg Lie algebra [22].The extension of the Heisenberg Lie algebra by this derivation has been used to classify Ricci solitons [23].Hogan and Lakey [24] studied extensions of the Heisenberg group in two ways.The first is the affine Weyl-Heisenberg (AWH) group, which is the semi-direct product of the Heisenberg group by dilations.Another way of extending the Heisenberg group is via the group of third-order upper triangular matrices, the so-called upper triangular (UT) groups.The UT and AWH groups are very closely related.In fact, they have isomorphic Lie algebras [24].The matrix realizations of these Lie algebras can be found in [23][24][25].

Filiform Lie Algebras
In 1983, Fialowski [26] has given a classification of infinite dimensional N-graded Lie algebras over a field of characteristic zero with two generators and one-dimensional homogeneous components.A Lie algebra of this type, satisfying [g 1 , g i ] = g i+1 for every i, must be isomorphic to one of the following: m 0 , m 2 or L 1 .They were called infinite-dimensional filiform Lie algebras as an infinite analogue of the finite-dimensional filiform Lie algebras coined by Vergne in [27].In this paper, we focus on m 0 , which has generators e i , i ≥ 1, and whose nontrivial relations are [e 1 , e i ] = e i+1 for all i ≥ 2. This Lie algebra has been studied by several authors, for instance, the cohomology of m 0 were computed in [28][29][30].We will also be dealing with a finite-dimensional filiform Lie algebra, called model filiform Lie algebra, whose nontrivial relations are given by [e 1 , e h ] = e h−1 for 3 ≤ h ≤ n [31] (see also [5]).

Carnot Algebras
Spaces with metric structure that is viewed as a constrained geometry, wherein motion is only allowed along a set of directions, changing from point to point, are called Sub-Riemmanian spaces, or Carnot-Caratheodory spaces [32].A Carnot algebra is a finitedimensional graded nilpotent Lie algebra, which is generated by the first homogeneous component and is equipped with a Euclidean structure.We call Carnot group the simply connected Lie group associated with a Carnot algebra.The Heisenberg group is the unique three-dimensional Carnot group, up to metric-preserving isomorphism [33].The graded automorphisms of Carnot algebras were studied in [34].

Engel-Type Algebras
Recently, Le Donne and Moisala [35] introduced the Engel-type algebra as a generalization of the four-dimensional Engel algebra [36].They are the only obstruction to semigeneration in Carnot algebras [35].The Betti numbers of Engel-type algebras were computed in [37], although Pouseele did not use the term Engel-type.Definition 2 ([35]).For each n ∈ N, the nth Engel-type algebra, denoted by E n , is the 2(n + 1)dimensional Lie algebra with basis {X, Y i , T i , Z} n i=1 , with nontrivial brackets given by The first Engel-type algebra E 1 , more popularly known as the Engel algebra, is the four-dimensional nilpotent Lie algebra denoted as n 4,1 in [5], and as N 4,2 in [38].Following the convention in [35,38], their commutators are illustrated by the diagram below where it is read from left to right of the V and the resulting bracket is at the bottom of the V. Here,

Z
The second one, which is six-dimensional, is denoted as E 2 .This can be found as N 6,3,1a in [38], as L 6,19(1) in [39], and as n 6,13 in [5] with = 1.The diagram below should be read similarly except that we follow the direction of the indicated arrows, for instance,

Solvable Lie Algebras with Engel-Type Nilradical
Rubin, Winternitz, and Ndogmo studied solvable extensions of a given nilpotent Lie algebra, such as the Heisenberg algebras [20], Abelian algebras [40], and so on (see [5]).Recently, Le Donne and Moisala [35] introduced the Engel-type algebras as a generalization of the Engel algebra.Here, we study solvable Lie algebras with Engel-type nilradical.This is a class of solvable Lie algebras that arises naturally in white noise analysis.As a starting point, a Lie algebra of white noise operators generated by annihilation and creation operators corresponding to elements of an orthonormal basis is isomorphic to the Heisenberg algebra.From here on, we assume that as defined in (4).Since φ e n k = 0, we must have α k = 0. Therefore, α k = 0 for all k.
Proof.We have the following correspondence a(e i ) → p i , a * (e i ) → q i , and Id → h.Moreover, [a(e i ), a * (e i )] = e i , e j Id = δ ij Id.The result follows from Theorem 4.
Then we extend the Heisenberg algebra H(n) by the Gross Laplacian ∆ G .However, this extension does not have the Heisenberg algebra as nilradical since it is nilpotent.We show that this is isomorphic to the Engel-type algebra.
Proof.We have the following correspondence − 1 2 ∆ G → X, a * (e i ) → Y i , a(e i ) → T i , and −Id → Z (see Section 2).The result follows from Theorem 4.

−Id
Finally, we extend the Engel-type algebra E n by the number operator.This we call a solvable Lie algebra with Engel-type nilradical.Few studies have been performed on this class of Lie algebras.A possible reason is that the number operator and Gross Laplacian differ only in infinite dimensions.In finite dimensions, they coincide (see (3)).An example can be found in [5] denoted as s 5,33 with nilradical n 4,1 .This has also been studied in [12,14] in terms of white noise operators.The following result can be easily shown, see [14] or [41] for proof of solvability and non-nilpotency.
Remark 2. As can be easily seen from Theorem 4, the number operator prevents nilpotency of h.Moreover, as a consequence of the canonical commutation relation of annihilation and creation operators, every nonzero ideal of h contains the identity (see [41]).Thus h cannot be semisimple; it cannot be written as direct sum of (simple) ideals.

Matrix Representation
A matrix representation of the extension of the Heisenberg Lie algebra by the number operator can be found in [25].However, this representation is limited since it cannot be generalized to include the Gross Laplacian.Fortunately, from Calvaruso's work in [23], we have a matrix representation of the five-dimensional Lie algebra in Theorem 8, which we reproduce here for convenience.

Characterization of Dimension
Given a topological vector space X over K = R or C, the set of all continuous linear operators on X is denoted by L(X).We define the iterates T n : X → X, n ≥ 0, by the following n-fold iteration of T, with T 0 = Id, where Id is the identity operator on X.If T ∈ L(X), we define the T-orbit of a vector x ∈ X to be the set Orb(x, T) = {x, Tx, T 2 x, . . .}.
See [42][43][44] for more details on orbits of operators.We are motivated by the quantum white noise derivatives of the conservation operator Λ(S).In particular, for S ∈ L(E) and ζ ∈ E, it is known that Our goal is to characterize the dimension of the Lie algebra It is obvious that if Orb(ζ, T) is finite, then h is finite dimensional.We will now deal with the case that Orb(ζ, T) is infinite.

Finite-Dimensional Case
Let us consider the case that T ∈ L(E) has eigenfunctions.For example, the Hermite functions are eigenfunctions of A = − d 2 dx 2 + x 2 + 1 [8] p. 17.We are interested in answering the question: given that T has eigenfunctions, which elements ζ ∈ E will make h in (20) finite dimensional?Lemma 3. Let T ∈ L(E) and x ∈ E. If Orb(x, T) is linearly dependent, then there exists n such that T m x ∈ span{T k x : k = 0, 1, . . ., n − 1} for all m ≥ n.
Proof.Suppose Orb(x, T) is linearly dependent.Then, there exists a number n such that By (21), we find that TV n ⊆ V n , since T i x ∈ V n , if i < n − 1 by definition.The lemma is proved.
Proof.By previous lemma, there exists n such that T m x ∈ span(T k x : k = 0, 1, . . ., n − 1) for all m ≥ n.Suppose m ≥ n.Then there are a k ∈ K such that Hence for all m ≥ n.
Remark 3. If x is an eigenfunction of T, then T n x = λ n x for all n.Thus, Orb(x, T) is linearly dependent.
In [41], we noted that h is finite dimensional if S is a finite rank operator.Here we provide a proof.
Proof.We have that dim K T(E) = n < ∞.Hence for every x ∈ E we find that Orb(x, T) ⊆ Kx + T(E) and dim K (Kx + T(E)) ≤ n + 1.Then Orb(x, T) is linearly dependent and we can apply Lemma 4.
Hence for every x ∈ V, we obtain Orb(x, T) ⊆ Kx + V.By similar arguments with the previous proof, the set {x, Tx, . . ., T n x} is linearly dependent.Therefore, h is finite dimensional.
The leftmost matrix is a Vandermonde matrix whose determinant is nonzero since the λ i 's are distinct.Thus it is invertible and b i = 0 for all i.This shows that {x, Tx, . . ., T n−1 x} is linearly independent but by previous proposition {x, Tx, . . ., T n x} is not.

Filiform Lie Algebras
By introducing a grading to Lie algebras of white noise operators, we establish isomorphism to filiform Lie algebras.The following result is due to Vergne [27] which was translated in [28].

Theorem 10 ([27]
).Let g = i≥1 g i be a N-graded Lie algebra with the following properties Then g is isomorphic to m 0 .Suppose h in ( 20) is infinite dimensional.We introduce a grading on h as follows: where h i = K f i for i ≥ 1.The basis elements f i are given by Applying the previous Theorem, Lemma 2, and Proposition 2, we obtain the following result.
Theorem 11.Suppose {e n } ∞ n=0 ⊂ E is an orthonormal basis for H.If S is the unilateral shift, then the Lie algebra Λ(S), a * (e k ) : k ≥ 0 is isomorphic to the infinite dimensional filiform Lie algebra m 0 .
It is known that the adjoint of the unilateral shift S is the backward shift S * which is characterized by S * e 0 = 0 and S * e n = e n−1 for n ≥ 1 [46].Let Hence we obtain the following result.
Theorem 12. Suppose {e n } ∞ n=0 ⊂ E is an orthonormal basis for H.If S is the unilateral shift, then the Lie algebra Λ(S), a(e 0 ), . . ., a(e n−1 ) is isomorphic to the (n + 1)-dimensional model filiform Lie algebra.

Solvable Lie Algebras with Abelian Nilradical
Solvable Lie algebras with Abelian nilradical are important in applications involving partial differential equations.A Lie algebra containing an Abelian subalgebra often arises because of equations with constant coefficients [40].They also arise as Lie algebras of vector groups and gauge groups [2].Examples of Abelian Lie algebras in white noise analysis include Lie algebras spanned by annihilation operators or those spanned by creation operators [8,11].Furthermore, any two generalized Gross Laplacians or any two conservation operators of diagonal type commute with each other [10].
In finite dimensional case, it is difficult to classify h in (20) without knowing the properties of the operator T.Here we deal with a solvable Lie algebra with one nonnilpotent element and an Abelian nilradical (see 10.4 [5]).Consider the following Lie algebra denoted as s 5,9 in [5] with nontrivial brackets given below.The values of the parameters a, b, c are Remark 5 ([47]).Suppose T is an orthogonal transformation on E. If a is an eigenvalue for T, then necessarily |a| = 1.

A solvable Lie Algebra with Abelian Nilradical sa n+1
We remark that s 5,9 is part of a family of solvable Lie algebras with Abelian nilradical such as s 3,1 , s 4,3 , and s 6,17 (see [5]).We define its generalization as follows.Denote by sa n+1 the Lie algebra with nontrivial brackets given below.The values of the parameters a i are If one or more equalities hold, further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in [5] 10.4.For our next result, we need the following commutation relation from [14].For κ ∈ (E ⊗m C ) * , [N, Ξ 0,m (κ)] = −mΞ 0,m (κ).

The Infinite Dimensional Unitary Group
Before we define the unitary group, let us first discuss its importance and applications.Let E denote a nuclear space which is dense in L 2 (R d ).The rotation group of E, denoted by O(E), keeps the white noise measure invariant via its adjoint action.It is known that a subgroup of O(E), called the conformal group C(d) generated by shifts, dilations, rotations and special conformal transformations, is isomorphic to the Lie group SO(d + 1, 1) [48].The diffeomorphisms of the parameter space R is a subclass of O(E), called Class II [49].There exists a three-dimensional subgroup of class II which describes the projective invariance of Brownian motion.Their generators span a three-dimensional Lie algebra isomorphic to the special linear Lie algebra sl(2, R).Perhaps most important is the generator of the shift which corresponds to the flow of Brownian motion.The rotation group O(E) may be regarded as a subgroup of the unitary group U(E C ) [50], which we now define below.
Let us denote by U(E C ) the set of all linear transformations g on E C that satisfy the following properties: 1.
g is a homeomorphism of E C onto itself, and 2.
g preserves the L It can be easily seen that U(E C ) is a group under the following product: We then call U(E C ) the infinite dimensional unitary group, or just the unitary group.

The Unitary Group U(S C )
In what follows, we take E C to be the complex Schwartz space S C .The structure of S C allows us to study the properties of U(S C ) in more detail.The case E C = S C is emphasized because the Fourier transform plays a significant role.Recall from Section 2.1.1 that the differential operator D = d 2 du 2 − (u 2 + 1) has the Hermite functions as its eigenfunctions with eigenvalues −2(n + 1): In fact, {e n : n ≥ 0} form an orthonormal basis for L 2 (R).
Proposition 6 ([50]).When considered as linear operators on S C , the Fourier transform F and the inverse Fourier transform F −1 (= F * ) belong to U(S C ).

The Group of Fourier-Mehler Transforms
In [50], Hida investigated a one-parameter subgroup of U(S C ), which consists of powers of the Fourier transform, not necessarily integral.Start with an integral kernel This gives us a one-parameter group {F θ : −∞ < θ < ∞} with period 2π.We recover the usual Fourier transform and its inverse by setting and we hence have obtained a one-parameter unitary group including them.For r ∈ R, the formal identity holds F πr/2 = (F ) r .
In this way, we may regard {F θ } as the subgroup of U(S C ) consisting of arbitrary powers of the Fourier transform.
where each ζ k is nonzero and belongs to the eigenspace of e ikθ .If h is an (n + 1)-dimensional complex Lie algebra spanned by then h ∼ = sa n+1 with parameters e ikθ , k = 1, . . ., n − 1.
Theorem 15.If S is the unilateral shift, then {∆ G (S k ) : k ≥ 0} is linearly independent.
As with annihilation operators, quadratic annihilation operators commute with each other [10,11].We have the following corollary.

Conclusions
We studied Lie algebras of white noise operators containing the quantum white noise derivatives of the conservation operator corresponding to a continuous linear operator S on Schwartz space.The dimension is characterized by the S-orbit of the initial vector ζ.In [35], Le Donne and Moisala introduced Engel-type algebras, which characterize semigeneration in Carnot algebras.We have shown that Engel-type algebras also arise in white analysis.In particular, the (2n + 3)-dimensional Lie algebra spanned by pairs of annihilation and creation operators, number operator and Gross Laplacian is a solvable Lie algebra with Engel-type nilradical.We have provided a matrix representation of this Lie algebra.
Furthermore, we have established isomorphisms between white noise Lie algebras containing a conversation operator and filiform Lie algebras.The Lie algebra spanned by the conservation operator corresponding to the unilateral shift and creation operators is isomorphic to the infinite dimensional filiform Lie algebra m 0 , while the Lie algebra spanned by the conservation operator and annihilation operators is isomorphic to the finite dimensional model filiform Lie algebra.
We have introduced a solvable Lie algebra with Abelian nilradical sa n+1 whose parameters have modulus less than or equal to one.For instance, Lie algebras of white noise operators spanned by the number operator and annihilation operators of any order belongs in this class.Moreover, the (n + 1)-dimensional complex Lie algebra spanned by the conservation operator corresponding to the Fourier-Mehler transform on the complex Schwartz space is isomorphic to sa n+1 with parameters nth roots of unity.We have shown the linear independence of quadratic annihilation operators corresponding to the nth powers of the unilateral shift-they form an Abelian Lie algebra.
Identifying which abstract Lie algebra a certain white noise Lie algebra is isomorphic to is valuable.For instance, a researcher who studies differential equations of white noise operators will benefit from knowing their invariants.The number operator and Gross In particular, a(ζ) and a * (ζ) are linear in ζ.
The analogous result holds for the creation operator.If {e n } ∞ 0 ⊂ E is an orthonormal basis for H, then {a(e n ) : n ≥ 0} and {a * (e n ) : n ≥ 0} are linearly independent sets.
n, we have