Integration of the lifting formulas and the cyclic homology of the algebras of differential operators (Q5956639)

Denote by \(\mathfrak{gl}_{\infty}^{\text{fin}}(\text{Dif}_n)\) the direct limit Lie algebra of finite matrices of differential operators on \(\mathbb{C}^n\). Feigin-Tsygan's theorem (in some form also attributed to Loday-Quillen) states that \[ H^{\bullet}_{\text{Lie}}(\mathfrak{gl}_{\infty}^{\text{fin}}(\text{Dif}_n))\cong \Lambda^{\bullet}(\Psi_{2n+1},\Psi_{2n+3},\Psi_{2n+5},\ldots).\tag{*} \] The main goal of the article under review is to show that the cocycles defined by the author in [\textit{B. Shoikhet}, Transl. Math. Monogr. 185, 95-110 (1998; Zbl 0928.17024); Math. Res. Lett. 6, 323-334 (1999; Zbl 0990.17018)] using lifting formulas can be taken as primitive generators of the Hopf algebra on the RHS of (*). The idea is to integrate the cocycles to cocycles on \(\mathfrak{gl}_{\infty}^{\text{fin}}(\text{Dif}_{\lambda,M})\) where \({\text{Dif}}_{\lambda,M}\) is the sheaf of holomorphic differential operators on an \(n\)-dimensional complex manifold \(M\) with values in a line bundle \(\lambda\). Here integration is meant in the sense of I. M. Gelfand and D. B. Fuks who showed that the Gelfand-Fuks cocycle (defining the universal central extension of the Lie algebra of vector fields on the circle \(S^1\)) is the integral over the circle of the Godbillon-Vey cocycle on the Lie algebra of formal vector fields in one formal variable. In the following, we review the content section-wise: The lifting formulas for cocycles of \(\mathfrak{gl}_{\infty}^{\text{fin}}(\text{Dif}_n)\) are carefully reviewed in the first section. Given \(k\) derivations \(D_1,\ldots,D_k\) on the Lie algebra \({\mathcal A}\) underlying an associative algebra \(A\) such that \[ [D_i,D_j]=\text{ad} Q_{ij}, \qquad \text{Alt}_{ijl} D_l(Q_{ij})=0, \] the author constructs a family of \((k+1)\)-, \((k+3)\)-, \((k+5)\)-cocycles. The leading term of the \((k+1)\)-cocycle is given by the formula \[ \Psi_{k+1}(A_1,\ldots,A_{k+1})=\text{Alt}_{A_1,\ldots,A_k} \text{Alt}_{D_1,\ldots,D_k} \text{Tr}(D_1A_1\ldots D_kA_k A_{k+1}). \] For \({\mathcal A}=\psi\text{Dif}_n\) the algebra of pseudo-differential operators on \((S^1)^n\), the \(k=2n\) derivations are \(\text{ad}(\ln x_1),\ldots,\text{ad}(\ln x_n),\text{ad}(\ln \partial_1),\ldots,\text{ad}(\ln \partial_n)\). The lifting formulas for \(\text{Dif}_n\) are then obtained by pullback with respect to the natural embedding. It follows a more explicit, combinatorial treatment of the lifting formulas. The second section treats integration in Lie algebra cohomology. Let \(M\) be a closed oriented differentiable manifold, \(W_n\) the Lie algebra of vector fields on \(\mathbb{R}^n\) and \(\text{Vect}(M)\) the Lie algebra of vector fields on \(M\). Integration associates to a cocycle \(\xi\in C^k_{\text{Lie}}(W_n;\mathbb{C})\) and a singular cycle \(\sigma\in C^{\text{sing}}_l(M,\mathbb{C})\) a cocycle \(\int_{\sigma}\xi\in C^{k-l}_{\text{Lie}}(\text{Vect}(M);\mathbb{C})\) such that triviality of \(\xi\) or \(\sigma\) implies triviality of \(\int_{\sigma}\xi\). The construction uses a point \(x\in M\) and a coordinate system \(\phi\) in \(x\) to induce a map \[ \phi_{\text{Vect}}:\text{Vect}(M)\to W_n \] and finally a map \[ \phi_{\text{Vect}}^*:C^{\bullet}_{\text{Lie}}(W_n;\mathbb{C})\to C^{\bullet}_{\text{Lie}}(\text{Vect}(M);\mathbb{C}). \] Choosing a coordinate system depending smoothly on \(x\), one constructs a family of cocycles \[ \xi(x)\in C^k_{\text{Lie}}(\text{Vect}(M);\mathbb{C}) \] which are all cohomologous to each other. (The author does not comment on the fact that the existence of such a coordinate system implies that \(M\) is parallelizable.) Therefore, there exists a \(\Theta_1\in\Omega^1_M\otimes C^{k-1}_{\text{Lie}}(\text{Vect}(M);\mathbb{C})\) such that \[ d_{DR}\xi(x)=\delta_{\text{Lie}}\Theta_1. \] Inductively, one obtains \(\Theta_i\) such that \[ d_{DR}\Theta_i=\delta_{\text{Lie}}\Theta_{i+1}.\tag{**} \] The integral \(\int_{\sigma}\Theta_l\in C^{k-l}_{\text{Lie}}(\text{Vect}(M);\mathbb{C})\) is then a cocycle, because \[ \delta_{\text{Lie}}\int_{\sigma}\Theta_l=\int_{\sigma}d_{DR}\Theta_{l-1}=0. \] One example is the Gelfand-Fuks cocycle generating \(H^2_{\text{Lie}}(\text{Vect}(S^1);\mathbb{C})\) obtained by integrating the Godbillon-Vey cocycle \(\theta\) which generates \(H^3_{\text{Lie}}(W_1;\mathbb{C})\). The author then solves system (**) and shows that the constructed integrals satisfy the stated triviality condition. The third section is concerned with integration in the holomorphic setting. The main idea, due to B. Feigin, is here that cohomology of Lie algebras of holomorphic sheaves is understood as being the differential graded (d.g.) cohomology of the d.g. Lie algebra obtained from tensoring the sheaf with the Dolbeault resolution [cf. \textit{B. Feigin}, Proc. Int. Congr. Math., Kyoto, Japan 1990, Vol. I, 71-85 (1991; Zbl 0757.17016); \textit{F. Wagemann}, Commun. Math. Phys. 208, 521-540 (1999); Erratum 220, 453-454 (2001; Zbl 0980.17011)]. This scheme is applied to the sheaf of holomorphic differential operators \(\text{Dif}_{\lambda,M}\) in a line bundle \(\lambda\) on a holomorphic manifold \(M\). In case \(M=\mathbb{C}^n\), the formal lifting formulas can be extended to the steted d.g. Lie algebra . Then, the integration procedure is transposed to the holomorphic setting, integration over the fundamental class of a compact \(M\) in \(H^{\text{sing}}_{2n}(M;\mathbb{C})\) playing the rôle of a holomorphic non-commutative residue. The fourth section concerns computations for \(M=\mathbb{C}P^n\). The author shifts from the Dolbeault point of view to the Čech point of view in order to use the standard affine coordinate system \(\{U_i\}\) of \(\mathbb{C}P^n\) for explicit computations. The integral in Lie algebra cohomology is reconstructed in this setting. For this purpose, the author introduces some kind of realization \(\widetilde{\mathbb{C}P^n}\) of the simplicial space given by the thickened nerve of the covering \(\{U_i\}\). More explicitly, the pieces \(U_1,\ldots,U_{n+1},(U_i\cap U_j)\times\sigma^1,\ldots,(U_1\cap\ldots\cap U_{n+1})\times\sigma^n\) where \(\sigma^i\) is the standard \(i\)-simplex, are glued together such that \(U_i\) is glued to the \(0\)th face of \((U_i\cap U_j)\times\sigma^1\), \(U_j\) is glued to the \(1\)st face of \((U_i\cap U_j)\times\sigma^1\), and so on. With this construction at hand, explicit computations of the extended lifting formulas in the Čech setting are performed for \(\mathbb{C}P^n\). By means of these explicit formulas, the author shows that the cochains are non-zero on some cycle, showing that the integral is non-zero and showing a fortiori that the lifting classes for \(\text{Dif}_n\) are non-zero. Lastly, the fact that the lifting cocycles generate primitively \(H^{\bullet}_{\text{Lie}}(\mathfrak{gl}_{\infty}^{\text{fin}}(\text{Dif}_n);\mathbb{C})\) is shown in the \(n\)-dimensional setting. In conclusion, the article is rich in useful constructions and helps to clarify the situation around Feigin-Tsygan's difficult theorem, linking it, as in [\textit{B. Feigin} and \textit{B. Tsygan}, Suppl. Rend. Circ. Mat. Palermo (2) 21, 15-52 (1989; Zbl 0686.14007)], to index theory.