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To study $s$-homogeneous algebras, we introduce the category of quivers with $s$-homogeneous corelations and the category of $s$-homogeneous triples. We show that both of these categories are equivalent to the category of $s$-homogeneous algebras. We prove some properties of the elements of $s$-homogeneous triples and give some consequences for $s$-Koszul algebras. Then we discuss the relations between the $s$-Koszulity and the Hilbert series of $s$-homogeneous triples. We give some application of the obtained results to $s$-homogeneous algebras with simple zero component. We describe all $s$-Koszul algebras with one relation recovering the result of Berger and all $s$-Koszul algebras with one dimensional $s$-th component. We show that if the $s$-th Veronese ring of an $s$-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all $s$-homogeneous algebras with $s$-th Veronese rings ${bf k}langle x,yrangle/(xy,yx)$ and ${bf k}langle x,yrangle/(x^2,y^2)$. In particular, we show that all of these algebras are not $s$-Koszul while their $s$-homogeneous duals are $s$-Koszul.
In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs $(A,M)$, where $A
A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their app
In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly Kahler) manifold $mathbf{S}^3timesmathbf{S}^3$. First, we show that any Hopf hypers
We prove that for an irreducible representation $tau:GL(n)to GL(W)$, the associated homogeneous ${bf P}_k^n$-vector bundle $W_{tau}$ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in ${bf P}_k^n$, where $k$ is an
Each hypersurface of a nearly Kahler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $phi$. In this paper, we show that, in the homogeneous NK $mathbb{S}^6$