Related papers: List of Problems
Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We consider also a special categorical invariant $K_\Theta (H)$ of this geometry. The…
Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case…
Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover…
At the 2015 Workshop on Lefschetz Properties of Artinian Algebras, Junzo Watanabe conjectured that every graded Artinian complete intersection algebra with the standard grading can be embedded into another such algebra cut out by quadratic…
In this paper, first we construct a Lie 2-algebra associated to every Leibniz algebra via the skew-symmetrization. Furthermore, we introduce the notion of the naive representation for a Leibniz algebra in order to realize the abstract…
In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate…
We show that the combinatorial Lefschetz number is a topological invariant. This is an important result in itself; in order to point it out, we will also work here several relevant consequences in different directions. The first of them is…
The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly…
This introduction to Lifshitz-type field theories reviews some of its aspects in Particle Physics. Attractive features of these models are described with different examples, as the improvement of graphs convergence, the introduction of new…
Consider ideals $I$ of the form \[ I=(x_1^2,\dots, x_n^2)+\mathrm{RLex}(x_ix_j) \] where $\mathrm{RLex}(x_ix_j)$ is the ideal generated by all the square-free monomials which are greater than or equal to $x_ix_j$ in the reverse…
The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well…
Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of…
This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the…
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold…
Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\, \longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We address…
An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.
Recently there has been a dramatic increase in the performance of recognition systems due to the introduction of deep architectures for representation learning and classification. However, the mathematical reasons for this success remain…
We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of…