Hopf Algebraic Structure of the (p,q)-Square Heizenberg White Noise Algebra

The two-parameter quantum deformations algebras based on the Fock representation for the two parameters deformed quantum oscillator algebra obtained in (Riahi et al., 2020a) and its connection with the Meixner classes given in a series of papers (Berezansky, 1968; Berezansky and Kondratiev, 2013; Barhoumi and Riahi, 2010) which found a lot of interesting applications in quantum probability. The Hopf algebraic structure problem stated below has led, in the past 30 years, to a multiplicity of new results in different fields of mathematics and physics. The theory of multiparameter quantum deformations of Lie algebras (Hu, 1999; Riahi et al., 2021), Lie bialgebras (Song and Su, 2006; Yue and Su, 2008), and quantization of Lie algebras (Chakrabarti and Jagannathan, 1991; Song et al., 2008; Su and Yuan, 2010) play an essential role in the quantum white noise literature. More precisely, the Fock representation of two parameters deformed commutation relation was first studied by (Riahi et al., 2021) by constructing an interacting Fock space Fp,q() as the space of representation.


Introduction
The two-parameter quantum deformations algebras based on the Fock representation for the two parameters deformed quantum oscillator algebra obtained in (Riahi et al., 2020a) and its connection with the Meixner classes given in a series of papers (Berezansky, 1968;Berezansky and Kondratiev, 2013;Barhoumi and Riahi, 2010) which found a lot of interesting applications in quantum probability.The Hopf algebraic structure problem stated below has led, in the past 30 years, to a multiplicity of new results in different fields of mathematics and physics.The theory of multiparameter quantum deformations of Lie algebras (Hu, 1999;Riahi et al., 2021), Lie bialgebras (Song and Su, 2006;Yue and Su, 2008), and quantization of Lie algebras (Chakrabarti and Jagannathan, 1991;Song et al., 2008;Su and Yuan, 2010) play an essential role in the quantum white noise literature.
More precisely, the Fock representation of two parameters deformed commutation relation was first studied by (Riahi et al., 2021) by constructing an interacting Fock space Fp,q() as the space of representation.
(p,q)-Deformed Square Heizenberg White Noise Algebra For p,q ∈ R such that 0 < q < p ≤ 1, the (p,q)-Heizenberg algebra p,q is generated by Υ, Υ + and N satisfying the following relations: , , , , , , where for α ∈  the action of α N on an element fn with n ∈  is given by: 00 ,.


The first example is the one-mode realization of p,q.
If we put: ,, ,, where, Mz is the multiplication operator by z ∈ , in the space of all finite linear combinations of fn = z n, and Dp,q is the (p,q)-derivative defined by: Then the action of the generators on the basis vector fn is obtained as follows: where, [n]p,q is the two parameters deformation of n ∈ N, i.e.:   and we conclude that the algebra generated by {1, Dp,q, Mz, N} gives a representation of p,q.The second example is the infinite-dimensional analog representation.Define the operator Tp,q on  ⊗n by: where, I(σ) and C(σ) denotes respectively the number of inversions and conversions of the permutation σ ∈ Sn.Now we put: , , 1, , .
equipped with the following scalar product: .
The (p,q)-Fock space denoted by Fp,q() is defined as: .
For, t ∈ , let ∂t and ∂t * be the pointwise annihilation and creation operators on Fp,q() given by: where, δt is the delta function at t and stands for the symmetric tensor product.For each ξ ∈ , we define the creation operator a * (ξ) and the annihilation operator a(ξ) on Fp,q() as follows: Proposition 2.2 Riahi et al. (2020b) The algebra generated by {1, a * (ξ), a(ξ), N} give a fock realization of the (p, q)-Heizenberg algebra, i.e.: where, N is the preservation operator, i.e.: .
Let ∂et be the operator defined by: It is known from (Accardi et al., 1999) that: where, c ∈  is an arbitrary constant.
Theorem 2.4 One can see that if p = 1 and q  1, the commutation relations (2.7) -(2.10) become: This shows that the (p,q)-square white noise algebra gives the square white noise Lie algebra when p = 1 and q → 1.
Hopf Structure of the (p,q)-Square Heizenberg White Noise Algebra Lemma 3.1 Let Ψ1, Ψ2, Ψ3, and Ψ4 be four non-zero continuous functions satisfying the following conditions: and for any non-zero continuous function f, define ∆(f(N)) by: Then there exists a unique homomorphism: : , , , , , ,

Proof
By direct calculations we have: Then by using (2.7) we obtain: Moreover, using a basis (ζk)k of , (2.5) and (2.6) give:
Hence, using (3.1) we get: and by similar calculations we obtain: Which gives the statement.

Proof
By using (3.4), we get: Thus, using a basis (ζk)k of , we obtain: and: Hence, if Ψ1 = Ψ2, we conclude that: Similarly, if Ψ3 = Ψ4, one can see that: and: Which completes the proof.

Lemma 3.4
If Ψ1Ψ3 = 1 and Ψ2Ψ4 = 1, then there exists a unique linear map S of Sl2,p,q() such that: where, f is a continuous non-zero arbitrary function.

Proof
Using (2.3), we get: Moreover, it is easy to see that: and 2c ξ, η.1 + (2 + q + p Nξη )[Nξη]p,q coincide on the basis (ζk)k of .Hence we conclude that: this proves that S preserves (2.7).similarly, we can easily verify that (2.8) (2.10) are also preserved by S. Thus there exist a homomorphism satisfying (3.8): : 1 1 p q p q S S S  Lemma 3.5 The following diagrams are commutative: p q p q m p q p q p q p q m p q p q Sl Sl S where: and σ:  −→ Sl2,p,q() such that: Proof Using (3.3), (3.4), and (3.5) we get: Suppose that Ψ1, Ψ2, Ψ3, and Ψ4 satisfy the following conditions: