Null Kähler Geometry and Isomonodromic Deformations

We construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi–Yau threefold. Using twistor methods we show that, in dimension four—where there is a connection with dispersionless integrability—the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

Modules which are invariant under nilpotents of their envelopes and covers

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with [Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

A Notion of Inversion Number Associated to Certain Quiver Flag Varieties

We define an algebraic variety $X(d,A)$ consisting of matrices whose rows and columns are partial flags. This is a smooth, projective variety, and we describe it as an iterated bundle of Grassmannian varieties. Moreover, we show that $X(d,A)$ has a cell decomposition, in which the cells are parametrized by certain matrices of sets and their dimensions are given by a notion of inversion number. On the other hand, we consider the Spaltenstein variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are affine bundles over certain varieties called $Y_T$, parametrized by semistandard tableaux $T$. We show that the varieties $Y_T$ are in fact isomorphic to varieties of the form $X(d,A)$. We deduce that each variety $Y_T$ has a cell decomposition, in which the cells are parametrized by certain row-increasing tableaux obtained by permuting the entries in the columns of $T$ and their dimensions are given by the inversion number recently defined by P. Drube for such row-increasing tableaux.

RELATIVITY GROUPOID INSTEAD OF RELATIVITY GROUP

In 1908, Minkowski [13] used space-like binary velocity-field of a medium, relative to an observer. In 1974, Hestenes introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector [7, 8]. We propose to consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of (ternary) relative velocities in isometric special relativity (loop structure). We consider an algebra of many time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of relativity groupoid is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative Frölicher–Richardson operator algebra. The Lorentz covariance is the cornerstone of physical theory. Observer-dependence within relativity groupoid, and the Lorentz-covariance within the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than Lorentz-invariant.