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On the cohomology of the Lie superalgebra of contact vector fields on \(S^{1|m}\). (English) Zbl 1191.17005

Let \(S^{1|m}\) denote the \((1,m)\)-dimensional supercircle with \(m\) odd variables \((\theta_1,\dots,\theta_m)\). The standard contact form on \(S^{1|m}\) is given by \[ \alpha_m\,=\,dx+\sum_{i=1}^m\theta_id\theta_i. \] The Lie superalgebra of contact vector fields \({\mathcal K}(m)\) is by definition \[ {\mathcal K}(m)\,=\,\{v\in{\mathrm V}{\mathrm e}{\mathrm c}{\mathrm t} (S^{1|m})\,|\,\exists\,F_v\in{\mathcal C}^{\infty}(S^{1|m}):\,L_v\alpha\,= \,F_v\alpha\}, \] where \(L_v\) denotes the Lie derivative.
On the other hand, the space \({\mathcal S}{\mathcal P}(m)\) of symbols of pseudodifferential operators on \(S^{1|m}\) is spanned by the formal series \[ A\,=\,\sum_{k=-M}^{\infty}\sum_{\varepsilon=(\varepsilon_1,\dots,\varepsilon_m)} a_{k,\varepsilon}(x,\theta)\xi^{-k}\bar{\theta}^{\varepsilon_1}\dots \bar{\theta}^{\varepsilon_m}, \] where \(\varepsilon_i\in\{0,1\}\), and \(\bar{\theta}_i\) is the symbol of \(\frac{\partial}{\partial\theta_i}\) in the same way as \(\xi\) is the symbol of \(\frac{\partial}{\partial x}\). \({\mathcal S}{\mathcal P}(m)\) carries a natural super Lie algebra structure, for which \({\mathcal K}(m)\) embeds as a subalgebra. This endows \({\mathcal S}{\mathcal P}(m)\) with a \({\mathcal K}(m)\)-module structure.
The main theorem is the computation of the first Lie superalgebra cohomology space \[ H^1({\mathcal K}(3),{\mathcal S}{\mathcal P}(3)), \] while the spaces \(H^1({\mathcal K}(m),{\mathcal S}{\mathcal P}(m))\) for \(m=1\) and \(m=2\) have been computed in earlier papers [the authors, Bull. Soc. R. Sci. Liège 72, No. 6, 365–375 (2003; Zbl 1055.17009), the authors and S. Omri, J. Nonlinear Math. Phys. 13, No. 1–4, 523–534 (2006; Zbl 1146.17016)]. The authors exhibit explicitly three generating cocycles.

MSC:

17B56 Cohomology of Lie (super)algebras
17B55 Homological methods in Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
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