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.