Finite-dimensional Algebra Research Articles

On non-counital Frobenius algebras

A Frobenius algebra is a finite-dimensional algebra [Formula: see text] which comes equipped with a coassociative, counital comultiplication map [Formula: see text] that is an [Formula: see text]-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map [Formula: see text] as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

Mutating signed -exceptional sequences

AbstractWe establish some properties of $\tau$ -exceptional sequences for finite-dimensional algebras. In an earlier paper, we established a bijection between the set of ordered support $\tau$ -tilting modules and the set of complete signed $\tau$ -exceptional sequences. We describe the action of the symmetric group on the latter induced by its natural action on the former. Similarly, we describe the effect on a $\tau$ -exceptional sequence obtained by mutating the corresponding ordered support $\tau$ -tilting module via a construction of Adachi-Iyama-Reiten.

Biderivations of complete Lie algebras

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra [Formula: see text] with [Formula: see text] and [Formula: see text], called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by Tang in [X. Tang, Biderivations of finite-dimensional complex simple Lie algebras, Linear Multilinear Algebra 66(2) (2018) 250–259] that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.

On unimodular module categories

Let C be a finite tensor category and M an exact left C-module category. We call M unimodular if the finite multitensor category RexC(M) of right exact C-module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Formula omitted] for cluster algebras from moduli spaces of G-local systems

For a finite-dimensional simple Lie algebra g admitting a non-trivial minuscule representation and a connected marked surface Σ with at least two marked points and no punctures, we prove that the cluster algebra Ag,Σ associated with the pair (g,Σ) coincides with the upper cluster algebra Ug,Σ. The proof is based on the fact that the function ring O(AG,Σ×) of the moduli space of decorated twisted G-local systems on Σ is generated by matrix coefficients of Wilson lines introduced in [23]. As an application, we prove that the Muller-type skein algebras Sg,Σ[∂−1][32,24,25] for g=sl2,sl3, or sp4 are isomorphic to the cluster algebras Ag,Σ.

-PERPENDICULAR WIDE SUBCATEGORIES

Abstract Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $ -tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $ -tilting reduction. This leads to a natural way to extend the definition of the “ $\tau $ -cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $ -tilting finite case and by Igusa–Todorov in the hereditary case.

Tensor product of finite dimensional nilpotent evolution algebras

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out.

Symmetries of the Energy–Momentum Tensor for Static Plane Symmetric Spacetimes

This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy.

A note on bireflectional elements of an algebra

A classical theorem of Wonenburger, Djokovič, Hoffmann and Paige [2, 3, 7] states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field F is algebraically closed with characteristic other than 2. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over F .

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

A new look at Lie algebras

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Fp-Projective periodicity

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat Proj-periodic module is projective, any fp-injective Inj-periodic module is injective, and any Cot-periodic module is cotorsion. It is also known that any pure PProj-periodic module is pure-projective and any pure PInj-periodic module is pure-injective. Generalizing a result of Šaroch and Št'ovíček, we show that every FpProj-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every FpProj-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure PProj-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.

Equivalence of categories of simple Lie algebras in positive characteristic

In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent.

On Some Classes of Bases in Finite-Dimensional Lie Algebras

On Some Classes of Bases in Finite-Dimensional Lie Algebras

Contact Lie systems: theory and applications

A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields V. We call V a Vessiot–Guldberg Lie algebra. We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden–Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot–Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated with examples having relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville type is developed.

Structure of primitive axial algebras

A “fusion law” is a collection of multiplication rules among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras are closely related to 3-transposition groups and vertex operator algebras.In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite-dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular, all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.

Algebra extensions and derived-discrete algebras

Let be an algebra homomorphism between finite dimensional algebras. We prove that if is split, the derived-discreteness of A implies the derived-discreteness of B; if is separable and the right A-module BA is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.

Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M is an endofunctor S on M together with a natural isomorphism Hom_(M,N)⁎≅Hom_(N,S(M)) for M,N∈M, where Hom_ is the internal Hom functor of M. In this paper, we discuss the case where C and M are the category of finite-dimensional left modules over a finite-dimensional Hopf algebra H and a finite-dimensional left H-comodule algebra L, respectively. We give an explicit description of the relative Serre functor of M and its twisted module structure in terms of the Frobenius structure of L. We also study pivotal structures on M and give some concrete examples.

Simple weight modules for Yangian Y (𝔰𝔩2)

Let [Formula: see text] be a finite-dimensional simple Lie algebra over [Formula: see text]. A [Formula: see text]-module is said to be weight if it is a weight [Formula: see text]-module. We give a complete classification of simple weight modules for [Formula: see text] which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical [Formula: see text]-representation theory, we show that there exists a class of [Formula: see text] irreducible modules which have uniformly two-dimensional weight spaces.

On radicals of Novikov algebras

We show that in a prime nonassociative Novikov algebra every nonzero ideal is non-associative. We prove that Baer (and Andrunakievich) radical and the largest left quasiregular ideal coincide in finite dimensional Novikov algebras over a field of characteristic 0 or algebraically closed field of odd characteristic. We show non-existence of right quasiregular radical in finite dimensional Novikov algebras.