Journal of Geometry and Physics最新文献

We explore transposed Poisson structures on the Lie algebra: the deformed twisted Schrödinger-Virasoro algebra $D\left(\lambda \right)$, as well as two Lie superalgebras: the super-BMS₃ algebra and the twisted $N=1$ Schrödinger-Neveu-Schwarz algebra. Initially, we demonstrate the absence of non-trivial transposed Poisson structures on the Lie algebra $D\left(\lambda \right)$ for $\lambda \ne 1$ and provide an example of a transposed Poisson algebra with associative and Lie parts isomorphic to the algebra of triadic extended Laurent polynomials and $D\left(1\right)$. Subsequently, we establish that the super-BMS₃ algebra possesses non-trivial $\frac{1}{2}$-superderivations but lacks a non-trivial transposed Poisson structure. Finally, we prove that the twisted N=1 Schrödinger-Neveu-Schwarz algebra does not have non-trivial $\frac{1}{2}$-superderivations and thus lacks non-trivial transposed Poisson structures.

We find a family of Lax representations with a non-removable parameter for the Euler equation of dynamics of an inviscid incompressible fluid in vorticity form on a two-dimensional Riemannian manifold.

We study the deformation theory of pseudo-group structure, and the purpose of this paper is to give a new simple proof of the existence theorem of pseudo-group structure under certain assumptions.

We prove that the group of strict contactomorphisms of the standard tight contact structure on the three-sphere deformation retracts to its unitary subgroup $U\left(2\right)$.

We classify superpotentials for the Hamiltonian system corresponding to the cohomogeneity one gradient Ricci soliton equations. Aside from recovering known examples of superpotentials for steady solitons, we find a new superpotential on a specific case of the Bérard Bergery–Calabi ansatz. The latter is used to obtain an explicit formula for a steady complete soliton with an equidistant family of hypersurfaces given by circle bundles over $S^2\times S^2$. There are no superpotentials in the non-steady case in dimensions greater than 2, even if polynomial coefficients are allowed. We also briefly discuss generalised first integrals and the limitations of some known methods of finding them.

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact $S^1$-manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

We study the existence of $\text{SU}{\left(2\right)}^2$-invariant $G_2$-instantons on $R^4\times S^3$ with the coclosed $G_2$-structures found on [1]. We find an explicit 1-parameter family of $\text{SU}{\left(2\right)}^3$-invariant $G_2$-instantons on the trivial bundle on $R^4\times S^3$ and study its “bubbling” behaviour. We prove the existence a 1-parameter family on the identity bundle. We also provide existence results for locally defined $\text{SU}{\left(2\right)}^2$-invariant $G_2$-instantons.

Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of invariant solutions; thus, it is crucial to obtain a classification of subgroups in order to have an optimal system of inequivalent solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite–dimensional Lie algebra uses inner automorphisms that are obtained by exponentiating the adjoint groups. In this paper, we present an effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite–dimensional Lie algebra abstractly assigned by means of its structure constants, or realized in terms of matrices or vector fields, or defined by a basis and the set of non-zero Lie brackets. The algorithm is implemented in the computer algebra system Wolfram Mathematica™; some meaningful and non-trivial examples are considered.

We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have a focus at the Kepler center, or have its center at the Hookian center respectively. When the particle hits the boundary it is ideally reflected according to the law of reflection. These systems are known to be integrable.

We describe the consecutive billiard orbits in terms of their foci. We show that the second foci of these orbits always lie on a circle in the Kepler case. In the Hooke case, we show that the foci of the orbits lie on a Cassini oval. For both systems we analyze the envelope of the directrices of the orbits as well.

The purpose of the present paper is to study cohomologies and non-abelian extensions of λ-differential pre-Lie algebras. First, we consider representations and cohomologies of λ-differential pre-Lie algebras. Next, we investigate non-abelian extensions and classify the non-abelian extensions in terms of non-abelian cohomology groups. Furthermore, we address the inducibility of a pair of automorphisms on non-abelian extensions and develop the Wells exact sequences in the context of λ-differential pre-Lie algebras. Finally, we discuss these results in the case of abelian extensions of λ-differential pre-Lie algebras.