Algebras of quantum groups as quantized Poisson manifolds (Q2779197)

The aim of this paper is the study of a very special type of Poisson manifold \((\mathbb{R}^4,\{,\})\) with a not constant rank of \(\{,\}\). The starting point of this study is the following result which will be published in a forthcoming paper by the author:NEWLINENEWLINENEWLINEDenoting by \(C^\infty (\mathbb{R}^4)= C^\infty (\mathbb{R}^4, \mathbb{R})\) the set of the smooth functions on \(\mathbb{R}^4\) with \(C^\infty\)-topology and by \(C^\infty(\mathbb{R}^4) [[\nu]]\) the direct product NEWLINE\[NEWLINEC^\infty (\mathbb{R}^4) [[\nu]]= \prod_{k\geq 0} \nu^kC^\infty (\mathbb{R}^4),NEWLINE\]NEWLINE there is a deformation quantization (*-product) of \(C^\infty (\mathbb{R}^4)\) such that the relations of the algebra of functions on the quantum \(2\times 2\) matrices described in [\textit{V. G. Drinfeld}, Quantum groups, Proc. I. C. M. '86, Berkeley Calif. Vol 1, 798-820 (1987; Zbl 0667.16003)] are satisfied.NEWLINENEWLINENEWLINEThus, the algebra \(C^\infty(\mathbb{R}^4)[[\nu]]\) has a twofold non-commutating aspect: the first one is a deformed noncommutative algebra structure of \(C^\infty(\mathbb{R}^4)\) and the second one is a coproduct which looks like the \(2\times 2\) matrix coalgebra.NEWLINENEWLINENEWLINEThe Poisson structure \(\{,\}\) on \(\mathbb{R}^4\) used in this note comes from the defining relations of this algebra; so, it is the quadratic Poisson structure defined on its generators by: NEWLINE\[NEWLINE\{x,u\}=xu,\;\{x,v\}=xv,\;\{x,y\}= 2uv,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\{u,v\}=0,\;\{u,y\}=uy, \quad \{v,y\}=vy,NEWLINE\]NEWLINE where \(x,u,v,y\) are the natural coordinate functions on \(\mathbb{R}^4\). The rank of \(\{ , \}\) takes both values 0 and 2.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].