Class Of Triangles Research Articles

Tilting objects in triangulated categories

Based on Beligiannis’s theory in [Beligiannis, A. (2000). Relative homological algebra and purity in triangulated categories. J. Algebra 227(1):268–361], we introduce and study -tilting objects in a triangulated category, where is a proper class of triangles. We show that each -tilting object cogenerates an -cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the -tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be -1-Gorenstein. Finally, we give a one to one correspondence between the class of -tilting objects and the class of tilting subcategories in a suitable functor category.

Proper resolutions and Gorensteinness in triangulated categories

Let $\mathcal {T}$ be a triangulated category with triangulation $\Delta $, $\xi \subseteq \Delta $ a proper class of triangles and $\mathcal {C}$ an additive full subcategory of $\mathcal {T}$. We provide a method for constructing a proper $\mathcal {C}(\xi )$-resolution (respectively, coproper $\mathcal {C}(\xi )$-coresolution) of one term in a triangle in $\xi $ from those of the other two terms. By using this construction, we show the stability of the Gorenstein category $\mathcal {GC}(\xi )$ in triangulated categories. Some applications are given.

Transfer of derived equivalences from subalgebras to endomorphism algebras

In this paper, we construct derived equivalences between subalgebras of some [Formula: see text]-Auslander–Yoneda algebras from a class of triangles in idempotent complete triangulated categories. The derived equivalences are obtained by transferring subalgebras induced by triangles to endomorphism algebras induced by approximation sequences.

Balance of Tate cohomology in triangulated categories

Let \(\mathcal {C}\) be a triangulated category and \(\mathcal {E}\) a proper class of triangles. We show that Tate cohomology in triangulated category is balanced, i.e. there is an isomorphism \(\widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {P}}(A, B)\cong \widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {I}}(A, B)\) for any integer \(i\in \mathbb {Z}\), where the first cohomology group is computed by complete \(\mathcal {E}\)-projective resolution for \(A\in \mathcal {C}\) and the second one is computed by complete \(\mathcal {E}\)-injective coresolution for \(B\in \mathcal {C}\). This improves the theorem proposed by J. Asadollahi and Sh. Salarian [J. Algebra 299, 480-502 (2006)].

A Note on Derived Equivalences for Φ-Green Algebras

In this note, we construct derived equivalences for Φ-Green algebras from a class of triangles in algebraic triangulated categories, where Φ is a finite admissible set of a set $\mathbb {N}$ of the natural numbers.

Notes on proper class of triangles

Let T be a triangulated category and ξ a proper class of triangles. Some basics properties and diagram lemmas are proved directly from the definition of ξ.

The Planar Three-Body Problem, Symmetries and Periodic Orbits

We study the planar three-body problem using the principal axes of inertia frame, determined by one Euler angle in the planar case. Three variables are needed in this frame: two distances R1 and R2, related with the principal moments of inertia on the plane of motion of the particles, and an auxiliary angle σ. We also connect these coordinates with the shape sphere of the similarity class of triangles and we give a geometric interpretation of the angle σ. We then write the Hamiltonian and equations of motion in these coordinates to find the symmetries of involution and symmetric periodic orbits. To exemplify this method, we calculate periodic orbits of a three-body problem with two or three equal masses.

A characterization of Gromov hyperbolicity of surfaces with variable negative curvature

In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature K≤ −k² < 0, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization of Gromov hyperbolicity for this kind of surfaces.

A new characterization of Gromov hiperbolicity for negatively curved surfaces

In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or, Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces.

Regularized Green’s function and group of reflections in a cavity with triangular cross section

For a certain class of triangles (with angles proportional to π/N, N ≥ 3) we formulate the image method by making use of the group GN generated by reflections with respect to three lines which form the triangle under consideration. A regularized Green’s function (which is employed in Casimir energy calculations) is obtained by classification of subgroups of GN and corresponding fixed points in the triangle.

Casimir energy inside a cavity with triangular cross section

For a certain class of triangles (with angles proportional to π/N, N⩾3) we formulate an image method by making use of the group GN generated by reflections with respect to the three lines which form the triangle under consideration. We formulate the regularization procedure by classification of subgroups of GN and corresponding fixed points in the triangle. We then also calculate Casimir energy for a cavity of infinite height with triangular cross section for scalar massless fields. More detailed calculation is given for odd N.

Shapes of tetrahedra

The equivalence classes of triangles and tetrahedra with respect to the group of the space dilatations and translations can be expressed by quaternions and ordered pairs of quaternions, respectively. These quaternions and ordered pairs of quaternions are called space shapes of triangles and shapes of tetrahedra. Using shapes, we discuss the similarity of two tetrahedra and obtain a common way for description of affine invariants of three collinear points and four coplanar points in the Euclidean space. We also examine a two-parameter set of tetrahedra with the same centroid.

All Triangles For Which sin⁡Aa Is a Constant

Solving the following problem, suggested by the law of sines, provides several exciting and satisfying moments of insight:What is the equivalence class of triangles for which (sin A)/a is a constant i