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Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…

We study pairs of finitely generated modules over a principal ideal domain and their corresponding matrix representations. We introduce equivalence relations for such pairs and determine invariants and canonical forms.

In this paper we study injective modules over universal enveloping algebras of finite-dimensional Lie algebras over fields of arbitrary characteristic. Most of our results are dealing with fields of prime characteristic but we also…

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of…

We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the…

In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the…

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…

We relate the geometry of the resonance varieties associated to a commutative differential graded algebra model of a space to the finiteness properties of the completions of its Alexander-type invariants. We also describe in simple…

For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant…

An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…

The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.

The algebraic geometry of a universal algebra $\mathbf{A}$ is defined as the collection of solution sets of term equations. Two algebras $\mathbf{A}_1$ and $\mathbf{A}_2$ are called algebraically equivalent if they have the same algebraic…

The cohomology algebra of the canonical bundle of a compact K\"ahler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order…

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…

The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods…

We investigate various homotopy invariant formulations of commutative algebra in the context of rational homotopy theory. The main subject is the complete intersection condition, where we show that a growth condition implies a structure…

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural…

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals…

We study almost complete intersections ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics---depth or relation type---have maximal or minimal values in the class. The focus is on those ideals that…

We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit…