A new approach to hom-Lie bialgebras

In this paper, we introduce a new definition of a hom-Lie bialgebra, which is equivalent to a Manin triple of hom-Lie algebras. We also introduce a notion of an $\mathcal O$-operator and then construct solutions of the classical hom-Yang-Baxter equation in terms of $\mathcal O$-operators and hom-left-symmetric algebras

Left-symmetric bialgebroids and their corresponding Manin triples

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1604.0014

Lie 2-bialgebras

In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras and symplectic Lie algebras.Comment: 22 page

Counterexamples to the quadrisecant approximation conjecture

A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self intersections while the quadrisecant approximation of the other knot is a knot with different knot type.Comment: 10 pages, 6 figure

Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras

This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are therefore called the local cocycle 3-Lie bialgebra and the double construction 3-Lie bialgebra. They can be regarded as suitable extensions of the well-known Lie bialgebra in the context of 3-Lie algebras, in two different directions. The local cocycle 3-Lie bialgebra is introduced to extend the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a ternary variation of the classical Yang-Baxter equation, called the 3-Lie classical Yang-Baxter equation, a ternary $\mathcal{O}$-operator and a 3-pre-Lie algebra is established. In particular, it is shown that solutions of the 3-Lie classical Yang-Baxter equation give (coboundary) local cocycle 3-Lie bialgebras, whereas 3-pre-Lie algebras give rise to solutions of the 3-Lie classical Yang-Baxter equation. The double construction 3-Lie bialgebra is introduced to extend to the 3-Lie algebra context the connection between Lie bialgebras and double constructions of Lie algebras. Their related Manin triples give a natural construction of pseudo-metric 3-Lie algebras with neutral signature. Moreover, the double construction 3-Lie bialgebra can be regarded as a special class of the local cocycle 3-Lie bialgebra. Explicit examples of double construction 3-Lie bialgebras are provided.Comment: 30 page

Pre-symplectic algebroids and their applications

In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. %We study three classes of pre-symplectic algebroids in detail. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated to a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi-Civita connection in the corresponding pseudo-Riemannian Lie algebroid.Comment: 22 page

Minimal surfaces in the three dimensional sphere with high symmetry

Using the Lawson's existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three dimensional sphere. These surfaces contain the Clifford torus, the Lawson's minimal surfaces, and seven new minimal surfaces with genera 9, 25, 49, 121, 121, 361 and 841. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three dimensional sphere.Comment: 24 pages, 9 figure

F-manifold algebras and deformation quantization via pre-Lie algebras

The notion of an F-manifold algebra is the underlying algebraic structure of an $F$-manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F-manifold algebras are the corresponding semi-classical limits. We study pre-Lie infinitesimal deformations and extension of pre-Lie n-deformation to pre-Lie (n+1)-deformation of a commutative associative algebra through the cohomology groups of pre-Lie algebras. We introduce the notions of pre-F-manifold algebras and dual pre-F-manifold algebras, and show that a pre-F-manifold algebra gives rise to an F-manifold algebra through the sub-adjacent associative algebra and the sub-adjacent Lie algebra. We use Rota-Baxter operators, more generally O-operators and average operators on F-manifold algebras to construct pre-F-manifold algebras and dual pre-F-manifold algebras.Comment: 22 page

On the evolution of a fossil disk around neutron stars originating from merging white dwarfs

Numerical simulations suggest that merging double white dwarfs (WDs) may produce a newborn neutron star surrounded by a fossil disk. We investigate the evolution of the fossil disk following the coalescence of double WDs. We demonstrate that the evolution can be mainly divided into four phases: the slim disk phase (with time $\lesssim$ 1 yr), the inner slim plus outer thin disk phase (\sim 10-\DP{6} yr), the thin disk phase (\sim \DP{2}-\DP{7} yr), and the inner advection-dominated accretion flow plus outer thin disk phase, given the initial disk mass \sim 0.05-0.5\,M_{\sun} and the disk formation time $10^{-3}-1$ s. Considering possible wind mass loss from the disk, we present both analytic formulae and numerically calculated results for the disk evolution, which is sensitive to the condition that determines the location of the outer disk radius. The systems are shown to be very bright in X-rays in the early phase, but quickly become transient within $\lesssim$ 100 yr, with peak luminosities decreasing with time. We suggest that they might account for part of the very faint X-ray transients around the Galactic center region, which generally require a very low mass transfer rate.Comment: 29 pages, 4 figures, accepted by Ap

Unifying neutron star sub-populations in the supernova fallback accretion model

We employ the supernova fallback disk model to simulate the spin evolution of isolated young neutron stars (NSs). We consider the submergence of the NS magnetic fields during the supercritical accretion stage and its succeeding reemergence. It is shown that the evolution of the spin periods and the magnetic fields in this model is able to account for the relatively weak magnetic fields of central compact objects and the measured braking indices of young pulsars. For a range of initial parameters, evolutionary links can be established among various kinds of NS sub-populations including magnetars, central compact objects and young pulsars. Thus, the diversity of young NSs could be unified in the framework of the supernova fallback accretion model.Comment: 16 pages, 7 figures, 9 Oct. 2018 accepted by RA