Nilpotent Structures Research Articles

One-sided nilpotent semicommutative properties of rings

In this paper, we study the semicommutative properties of rings with one-sided nilpotent structures. The concepts of left and right nilpotent semicommutative rings are defined and studied, which are a generalization of semicommutative rings. We show that the class of one-sided nilpotent semicommutative rings is strictly placed between the class of semicommutative rings and that of nil-semicommutative rings. As applications, we give some characterizations of semiprime rings from the point of view of left nilpotent semicommutative rings.

Canonical nilpotent structure under bounded Ricci curvature and Reifenberg local covering geometry over regular limits

It is known that a closed collapsed Riemannian [Formula: see text]-manifold [Formula: see text] of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger–Fukaya–Gromov with respect to a smoothed metric [Formula: see text]. We study the nilpotent structures over a regular limit space with optimal regularities that describe the collapsing of original metric [Formula: see text], and prove that they are uniquely determined up to a conjugation by diffeomorphisms with bi-Lipschitz constant almost [Formula: see text], and are equivalent to nilpotent structures arising from other nearby metrics [Formula: see text] with respect to [Formula: see text]’s sectional curvature bound.

Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

Book Review: Nilpotent structures in ergodic theory

Book Review: Nilpotent structures in ergodic theory

Towards general super Casimir equations for 4D mathcal{N}=1 SCFTs

Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for 4D mathcal{N}=1 SCFTs. We begin by finding the most general four-point function with zero U(1)R-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where some of the operators satisfy shortening conditions. Finally, we obtain the super Casimir equations for four point-functions which contain a chiral and an anti-chiral field. We solve the super Casimir equations by writing the superconformal blocks as a sum of several conformal blocks.

Stability of Pure Nilpotent Structures on Collapsed Manifolds

AbstractThe goal of this paper is to study the stability of pure nilpotent structures on an $n$-manifold for different collapsed metrics. We prove that if two metrics with bounded sectional curvature are $L_0$-bi-Lipschitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeomorphism, the underlying nilpotent Killing structures coincide with each other, or one is embedded into another as a subsheaf. It improves Cheeger–Fukaya–Gromov’s local compatibility of pure nilpotent Killing structures for one collapsed metric to two Lipschitz equivalent metrics. As an application, we prove that those pure nilpotent Killing structures constructed by various smoothing methods to a Lipschitz equivalent metric with bounded sectional curvature are uniquely determined by the original metric modulo a diffeomorphism.

Odd-dimensional Weyl and pseudo-Weyl spaces with additional tensor structures

Odd-dimensional Weyl and pseudo-Weyl spaces admitting almost contact, almost paracontact and nilpotent structures are considered in this work. The results are obtained by means of the apparatus of the prolonged covariant differentiation. A linear connection with torsion is constructed. With respect to this connection the prolonged covariant derivatives of the fundamental tensors of the Weyl and pseudo-Weyl spaces are found to be zero. The curvature tensor with respect to this connection is considered. MSC(2010): 53B05, 53B99. Keywords: composition, almost contact structure, almost paracontact structure, nilpotent structure, Weyl space, prolonged differentiation.

SU(3)-holonomy metrics from nilpotent Lie groups

One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gauge transformations, and study the flow induced on the variety of frames on a Lie algebra taken up to automorphisms. We classify the orbits of this flow for all hypo nilpotent structures, obtaining several families of cohomogeneity one metrics with holonomy contained in SU(3). We prove that these metrics cannot be extended to a complete metric, unless they are flat.

Collapsing Construction with Nilpotent Structures

A fundamental result concerning collapsed manifolds with bounded sectional curvature is the existence of compatible local nilpotent symmetry structures whose orbits capture all collapsed directions of the local geometry [CFG]. The underlying topological structure is called an N-structure of positive rank. We show that if a manifold M admits such an N-structure \({\mathcal{N}}\), then M admits a one-parameter family of metrics g∈ with curvature bounded in absolute value while injectivity radii and the diameters of \({\mathcal{N}}\)-orbits away from the singular set of \({\mathcal{N}}\) uniformly converge to zero as \(\epsilon \rightarrow 0\). Moreover, g∈ is \({\mathcal{N}}\)-invariant away from the singular set. This result extends collapsing results in [CG1], [Fu3] and [G].

Conjugate radius and isometry group of a manifold with negative Ricci curvature

It is known that the order of the isometry group on a compact Riemannian manifold with negative Ricci curvature is finite. We show by local nilpotent structures that a bound on the orders of the isometry groups exists depending only on the Ricci curvature, the conjugate radius and the diameter.

Codimension two linear varieties with nilpotent structures

Codimension two linear varieties with nilpotent structures

Nilpotent structures and invariant metrics on collapsed manifolds

Let Mn be a complete Riemannian manifold of bounded curvature, say IKI 0, we put Mn = W n(c) U Fn (c), where ?1n (e) consists of those points at which the injectivity radius of the exponential map is > e. The complementary set, &n (c) is called the e-collapsed part of Mn. If x E 4 (n), r < E, then the metric ball Bx (r) is quasi-isometric, with small distortion, to the flat ball B0(r) in the Euclidean space, Rn . After slightly

Solvable and Nilpotent Structures

In [6], we have proven that every non-Hj-categorical connected nilpotent group of finite Morley rank has the center Z(G) with RM(Z(G))>2. If a connected nilpoten group has a finite Morley rank, then it is easy to see that the center is infinite. So the above result shows if G is non-Ki-categorical the center has a rather complicated structure. The present paper is a natural continuation of [6]. In this paper we introduce the notion of *-nilpotency, and prove aversions of results in [6]. (See Theorems 2.3, and 2.6.) Since our setting is rather general, we can apply our results to <y-stable ring theory. (See Corollary 2.5.)