A universal deformation formula for \(\mathcal H_{1}\) without projectivity assumption
The authors aim to find an \textit{universal deformation formula} for Connes-Moscovici's Hopf subalgebra \({\mathcal H}^1_{CM}\subset {\mathcal H}_{CM}\), by using a Fedosov's quantization type for symplectic diffeomorphisms. By universal deformation formula of a Hopf algebra \(A\), they mean an element \(R\in A\hbar\bigotimes_{{\mathbb C}\hbar} A\hbar\) satisfying the following conditions: (a) \(((\triangle\otimes 1)R)(R\otimes 1)=((1\otimes\triangle)R)(1\otimes R)\); (b) \((\epsilon\otimes 1)(R)=1\otimes 1=(1\otimes\epsilon)(R)\). Here \(\triangle\) is the coproduct and \(\epsilon\) the counit in \(A\). The Connes-Moscovici Hopf algebra \({\mathcal H}_{CM}\) is the enveloping algebra of the Lie algebra which is the linear space \(<X, Y,\delta_n>_{n\geq 1}\), with the following brackets: \[ (\clubsuit)\quad[X,Y]=X,\quad [Y,\delta_n]=n\delta_n,\quad[\delta_n,\delta_m]=0, \quad[X,\delta_n]=\delta_{n+1},\quad m,n\geq 1. \] The coproduct \(\triangle\) in \({\mathcal H}_{CM}\) is defined by: \(\triangle(Y)=Y\otimes 1+1\otimes Y\), \(\triangle(X)=X\otimes 1+1\otimes X+\delta_1\otimes Y\), \(\triangle(\delta_1)=\delta_1\otimes 1+1\otimes \delta_1\), where \(\triangle(\delta_n)\) is defined recursively, using above brackets \((\clubsuit)\) and the identity \(\triangle(h_1h_2)=\triangle(h_1)\triangle(h_2)\), \(\forall h_1,h_2\in {\mathcal H}_{CM}\). The algebra generated by \(\{\delta_n\}_{n\geq 1}\) is the graded sub-Hopf algebra \({\mathcal H}^1_{CM}\subset {\mathcal H}_{CM}\), where the gradiation is defined by \(gr(\delta_{n_1},\dots,\delta_{n_s})=n_1+\dots+n_s\). \({\mathcal H}^1_{CM}\) is Hopf-isomorphic to the so-called \textit{de Fadd di Bruno algebra}, \({\mathcal H}_{FB}\equiv{\mathbb R}[a_1,\dots,a_n,\dots]\), where \(a_n:G\to{\mathbb R}\) are suitable functions over the group \(G\) of so-called \textit{formal tangent diffeomorphisms}. The \textit{Fedosov's quantization} of a symplectic manifold \((M,\omega)\) is obtained by a bijection \(f\mapsto\hat f\), between the commutative algebra \(C^\infty(M)\) and a noncommutative algebra \({\mathcal W}_D\), identified by a connection \(D\), (\textit{Fedosov's connection}), on a suitable vector fiber bundle \(W\to M\), depending on a \textit{deformation formal parameter} \(\hbar\). The product in \({\mathcal W}_D\) can be considered a generalized ``Moyal product''. This type of quantization can be extended to symplectic diffeomorphisms by extending the fiber bundle \(W\). Then by generalizing the Fedosov's method they are able to obtain a quantization of a pseudogroup \(\Gamma\) of a symplectic manifold \(M\). Their main result is to prove that, by considering \(M\equiv {\mathbb R}\times {\mathbb R}^+\), it is possible construct a Moyal type product on \(C^\infty_0(M)\times_r\Gamma\hbar\) that can be expressed by an element \(R\in{\mathcal H}^1_{CM}\bigotimes_{{\mathbb C}\hbar}{\mathcal H}^1_{CM}\). (Here \(\times_r\) denotes semidirect product.) Reviewer's remark. Hopf algebras are very useful structures whether in commutative geometry or in noncommutative ones. Let us recall here that conservation laws of PDE's identify extended Hopf algebras that are commutative when PDE's are defined in the category of commutative manifolds, and result, instead, quantum algebras for quantum PDE's. Then extended (quantum) Hopf algebras play a central role to characterize global solutions in (quantum) PDE's. (See recent works on the geometry of (quantum) PDE's by the reviewer of this paper.)
scientific article; zbMATH DE number 5550886