On Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures. (English) Zbl 1389.53054

Janyška, Josef

Arch. Math., Brno 52, No. 5, 325-339 (2016).

The so-called generators of infinitesimal symmetries of almost-cosymplectic-contact structures \((\omega ,\Omega)\) on an odd-dimensional manifold \(M\) form Lie algebras. Some properties of these Lie algebras are considered. In particular, the Lie singularities in these Lie algebras are described.
The main result of the work is the following
Theorem. Let \(X\) be an infinitesimal symmetry of the almost-cosymplectic-contact structure \((\omega ,\Omega)\). Then the operator \(L_X\) is the derivation of the Lie algebra \(\left\langle \mathrm{Gen}(\omega ,\Omega),[,]\right\rangle\) of generators of infinitesimal symmetries of the structure \((\omega ,\Omega)\).
Here the differential operator \(L_x:\Omega^1(M)\times C^\infty (M)\rightarrow\Omega^1(M)\times C^\infty (M)\) is given by the Lie derivatives with respect to a vector field \(X\): \[ L_X(\alpha ,h)=(L_X\alpha ,L_Xh). \]

Reviewer: Mihail Banaru (Smolensk)

MSC:

53C15

General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

almost-cosymplectic-contact structure; Lie algebra; infinitesimal symmetry; Reeb vector field; derivation; almost-coPoisson-Jacobi structure