Related papers: Further results on modularity in evolution algebra…
In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in case of…
We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…
We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…
In this paper we classify extensions between irreducible finite conformal modules over the Virasoro algebra, over the current algebras and over their semidirect sums.
We develop the theory of the higher commutator for Taylor varieties. A new higher commutator operation called the hypercommutator is defined using a type of invariant relation called a higher dimensional congruence. The hypercommutator is…
In this paper, we introduce Volterra evolution algebras which are evolution algebras whose structural matrices are described by skew symmetric matrices. A main result of the present paper gives a connection between such kind of algebras…
We consider here the class of fully-nonlinear symmetry-integrable third-order evolution equations in 1+1 dimensions that were proposed recently in the journal Open Communications in Nonlinear Mathematical Physics, vol. 2, 216--228 (2022).…
Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad. The associated notion of morphism of left modules ("Linear" natural transformations) captures an important…
We review the new approach to the theory of nonlinear $W$-algebras which is developed recently and called {\it conformal linearization}. In this approach $W$-algebras are embedded as subalgebras into some {\it linear conformal} algebras…
We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different.…
We investigate the selective forces that promote the emergence of modularity in nature. We demonstrate the spontaneous emergence of modularity in a population of individuals that evolve in a changing environment. We show that the level of…
The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $\Lambda$ are explored, with regard to both their geometry and the structure of the modules they encode. Provided…
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…
Transfer learning has recently become the dominant paradigm of machine learning. Pre-trained models fine-tuned for downstream tasks achieve better performance with fewer labelled examples. Nonetheless, it remains unclear how to develop…
Modularity has been widely studied as a mechanism to improve the capabilities of neural networks through various techniques such as hand-crafted modular architectures and automatic approaches. While these methods have sometimes shown…
We show that the theories of some (ordered) central simple algebras with involution over real closed fields are model-complete or admit quantifier elimination, and characterize positive cones in terms of morphisms into models of some of…
We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: it can be explicitly described as a polynomial on its evolution parameter. Such a property is established…
We review the known results about characteristically nilpotent complex Lie algebras, as well as we comment recent developements in the theory.
In this paper we give classification of two-dimensional real evolution algebras. For several chains of evolution algebras we study their classification dynamics.