4,455 research outputs found

### 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

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