The group of volume preserving diffeomorphisms and the Lie algebra of unimodular vector fields: survey of some classical and not-so-classical results. (English) Zbl 1179.17020

Ali, S. Twareque (ed.) et al., Twenty years of Bialowieza: A mathematical anthology. Aspects of differential geometric methods in physics. Hackensack, NJ: World Scientific (ISBN 981-256-146-3/hbk). World Scientific Monograph Series in Mathematics 8, 79-98 (2005).
This survey reports on properties of Lie algebras of vector fields with vanishing divergence and their corresponding Fréchet Lie groups which are important for deformation quantization, namely cohomology, central extensions and coadjoint orbits.
Let \(V\) be an oriented manifold of dimension \(n\) and \(\omega\) a volume form. Vector fields \(X\) on \(V\) such that the divergence, defined by \(L_X\omega=d(i_X\omega)=\mathrm {Div}(X)\omega\), vanishes, are called unimodular. In case \(i_X\omega\) is not only closed, but exact, \(X\) is called exact unimodular. The Lie algebra of unimodular fields is denoted by \({\mathrm S}\overline{\mathrm{Vect}}(V)\), while the one of exact unimodular fields is written \(\mathrm{S}\mathrm{Vect}(V)\).
After a first section concerned with definitions, the author recalls in the second section A. Lichnerowicz’s results [Ann. Inst. Fourier 24, No. 3, 219–266 (1974; Zbl 0289.58002)] on derivations of the Lie algebra of unimodular and exact unimodular vector fields, emphasizing the short exact sequence \[ 0\to{\mathrm S}{\mathrm {Vect}}(V)\to{\mathrm S}\overline{{\mathrm {Vect}}}(V)\to H^{n-1}_{\mathrm dR}(V)\to 0. \] Then, the Gerstenhaber algebra and Batalin-Vilkovisky (BV) algebra structure of the graded algebra of contravariant anti-symmetric tensor fields \(\Omega_*(V)\) are reviewed.
The third section recalls P. B. A. Lecomte and C. Roger’s result on the rigidity of \({\mathrm S}\overline{{\mathrm {Vect}}}(V)\) [J. Differ. Geom. 44, No. 3, 529–549 (1996; Zbl 0869.17019)]. It states that \({\mathrm S}\overline{{\mathrm {Vect}}}(V)\) has no infinitesimal deformations if \(n\geq 4\), and that for \(n=3\), the space of infinitesimal deformations is \(1\) dimensional, but the obstructions to prolongation never vanish.
In a fourth section, the supergeometric variant of the preceding set up is discussed. Lichnerowicz’s results generalize to the supercase, and one can even get hold on the central extensions.
While the Lie algebra \({\mathrm S}\overline{{\mathrm {Vect}}}(V)\) integrates in a straight forward way to a Fréchet Lie group \({\mathcal S}{\mathcal D}(V)\) of volume preserving diffeomorphisms (with the usual difficulties of the infinite dimensional setting), the fifth section discusses how the above short exact sequence can be lifted to group level using the Calabi invariant.
The sixth section reviews R. S. Ismagilov’s work [Representations of infinite-dimensional groups. Translations of Mathematical Monographs. 152. Providence, RI: American Mathematical Society (1996; Zbl 0856.22001)] on the coadjoint orbits of \({\mathcal S}{\mathcal D}(V)\) in the regular dual of \({\mathrm S}{\mathrm {Vect}}(V)\).
The last paragraph gives perspectives of how to define deformation quantization of \({\mathrm S}\overline{{\mathrm {Vect}}}(V)\) inspite of its rigidity, and on links to the physics of branes.
Related work by other authors includes [C. Vizman, J. Lie Theory 16, No. 2, 297–309 (2006; Zbl 1128.17017)] and [K. H. Neeb, J. Geom. Symmetry Phys. 5, 48–74 (2006; Zbl 1105.53064)].
For the entire collection see [Zbl 1089.53002].


17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
17B65 Infinite-dimensional Lie (super)algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
53D55 Deformation quantization, star products
58A50 Supermanifolds and graded manifolds