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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 $A$ is the $s$-Veronese ring and $M$ is the $(s,1)$-Veronese bimodule of the $s$-homogeneous dual algebra. For each such a pair we give an intrinsic characterization of algebras corresponding to it. Due to results of our previous work many pairs determine the algebra uniquely up to isomorphism. Using our partial classification, we show that, to check the $s$-Koszulity of a connected $s$-homogeneous algebras with two relations, it is enough to verify an equality for Hilbert series or to check the exactness of the generalized Koszul complex in the second term. For each pair $(A,M)$ not belonging to one specific series of pairs, we check if there exists an $s$-Koszulity algebra corresponding to it. Thus, we describe a class of possible ${rm Ext}$-algebras of $s$-Koszul connected algebras with two relations and realize all of them except a finite number of specific algebras as ${rm Ext}$-algebras. Another result that follows from our classification is that an $s$-homogeneous algebra with two dimensional $s$-th component cannot be $s$-Koszul for $s>2$.
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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.
We describe the derived Picard groups and two-term silting complexes for quasi-hereditary algebras with two simple modules. We also describe by quivers with relations all algebras derived equivalent to a quasi-hereditary algebra with two simple modules.
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 application to gravity theories. It is conjectured that a CSI spacetime is either locally homogeneous or belongs to the subclass of degenerate Kundt metrics. Independent of this conjecture, any CSI spacetime can be related to a particular locally homogeneous degenerate Kundt metric sharing the same scalar polynomial curvature invariants. In this paper we will invariantly classify the entire subclass of locally homogeneous CSI Kundt spacetimes which are of alignment type {bf D} to all orders and show that any other CSI Kundt metric can be constructed from them.
The elliptic algebras in the title are connected graded $mathbb{C}$-algebras, denoted $Q_{n,k}(E,tau)$, depending on a pair of relatively prime integers $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},sigma,mathcal{L}_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,tau)$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ we show the homomorphism $Q_{n,k}(E,tau) to B(X_{n/k},sigma,mathcal{L}_{n/k})$ is surjective, that the relations for $B(X_{n/k},sigma,mathcal{L}_{n/k})$ are generated in degrees $le 3$, and the non-commutative scheme $mathrm{Proj}_{nc}(Q_{n,k}(E,tau))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$, respectively. When $X_{n/k}=E^g$ and $tau=0$, the results about $B(X_{n/k},sigma,mathcal{L}_{n/k})$ show that the morphism $Phi_{|mathcal{L}_{n/k}|}:E^g to mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.
We investigate maximal tori in the Hochschild cohomology Lie algebra $HH^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$, extending work of Farkas, Green and Marcos; de la Pe~na and Saorin; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.
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