Finite-dimensional Algebra Research Articles

Lifting and restricting t‐structures

AbstractWe explore the interplay between t‐structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring using homotopy colimits. More precisely, we show that every intermediate t‐structure in can be lifted to a compactly generated t‐structure in , by closing the aisle and the coaisle of the t‐structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t‐structure in to restrict to an intermediate t‐structure in , thus describing which t‐structures can be obtained via lifting. We apply our results to the special case of HRS‐t‐structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.

Almost split sequences in tri-exact categories

We study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend a number of existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories by numerous researchers. As applications, we obtain some new results on the existence of almost split triangles in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.

Remarks on Ultrasemiprime algebras

Every finite dimensional normed algebra is isomorphic to the finite direct product of or , it is also proved these algebras are ultrasemiprime algebras. In this paper, the ultrasemiprime proof of the finite direct product of and is generalized to the finite direct product of any ultrasemiprime algebras.

Generalization of Jordan-Lie inner ideals of finite dimensional associative algebras

We generalize Baranov and Shlaka’s results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let [Formula: see text] be a finite dimensional [Formula: see text]-perfect associative algebras [Formula: see text] over an algebraically closed field [Formula: see text] of arbitrary characteristic [Formula: see text] and let [Formula: see text] be a subalgebra of [Formula: see text]. We prove that for any bar-minimal Jordan-Lie inner ideal [Formula: see text] a [Formula: see text], there is a bar-minimal Jordan-Lie inner ideal of [Formula: see text] that contains [Formula: see text] and if [Formula: see text] is regular, then is a regular inner ideal of [Formula: see text] that contains [Formula: see text]. We also prove that for any strict orthogonal pair [Formula: see text] in [Formula: see text], there is a strict orthogonal idempotent pair [Formula: see text] in [Formula: see text] such that [Formula: see text].

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Dense Orbits of a Two Point Algebra

Finite dimensional algebras over a field can be classified into three classes by representation type. These are finite, tame, and wild. Wild algebras contain copies of the representations of all finite dimensional algebras so classifying the indecomposable representations is not feasible. We then look at representations whose orbit is dense in their associated irreducible component of their representation variety. With this geometric perspective, we can approximate our representations. We give an example of a two point dense orbit algebra of wild representation type and show that it has a dense orbit for dimension vectors of certain types.

Representations of weakly triangular categories

A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of a split triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined so as to give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration.As an application, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras. Finally, we obtain categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or slˆe.

Crossed products of 4-algebras. Applications

A 4-algebra is a commutative algebra A over a field k such that (a2)2=0, for all a∈A. We have proved recently [22] that 4-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let A be a 4-algebra, E a vector space and π:E→A a surjective linear map with V=Ker(π). All 4-algebra structures on E such that π:E→A is an algebra map are described and classified by a global cohomological object GH2(A,V). Any such 4-algebra is isomorphic to a crossed product V#A and GH2(A,V) is a coproduct, over all 4-algebras structures ⋅V on V, of all non-abelian cohomologies Hnab2(A,(V,⋅V)), which are the classifying objects for all extensions of A by V. Several applications and examples are provided: in particular, GH2(A,k) and GH2(k,V) are explicitly computed and the Galois group Gal(V#A/V) of the extension V↪V#A is described.

Levi Decomposition of Frobenius Lie Algebra of Dimension 6

In this paper, we study notion of the Lie algebra of dimension 6. The finite dimensional Lie algebra can be expressed in terms of decomposition between Levi subalgebra and the maximal solvable ideal. This form of decomposition is called Levi decomposition. The work aims to obtain Levi decomposition of Frobenius Lie algebra of dimension 6. To achieve this aim, we compute Levi subalgebra and the maximal solvable ideal (radical) of with respect to its basis. To obtain Levi subalgebra and the maximal solvable ideal, we apply literature reviews about Lie algebra and decomposition Levi in Dagli result. For future research, decomposition Levi for higher dimension of Frobenius Lie algebra is still an open problem.

On properties of some nilpotent evolution algebras and their enveloping algebras

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.

Homotopies of Constant Cuntz Class

Abstract Let $A$ be a unital simple separable exact C$^{\ast }$-algebra that is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed Cuntz class is path connected. This result applies in particular to irrational rotation algebras and approximately finite-dimensional (AF) algebras.

On existence of PI-exponent of algebras with involution

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of ⁎-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional ⁎-PI-exponent. We also construct a family of infinite-dimensional algebras Cα such that exp⁎(Cα) does not exist.

Two-sided zero product properties on symmetric algebras

We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ(x,y)=0 whenever xy=yx=0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x,y)↦τ1(xy)+τ2(yx) for x,y∈A, where τ1,τ2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A, there exists a∈A such that, for all x∈A, d(x)−[a,x] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd.

A Friendly Introduction to Path Algebras and Quiver Representations

We give an elementary introduction to the theory of quiver representations, which is the foundation of a dynamic area of research. Relying only on basic concepts from linear and abstract algebra, we construct the path algebra of a quiver, and illustrate the notion of the representation type of a finite-dimensional algebra.

Classification of finite-dimensional Hopf algebras over dual Radford algebras

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.

Reduction techniques of singular equivalences

It is shown that a singular equivalence induced by tensoring with a suitable complex of bimodules defines a singular equivalence of Morita type with level, in the sense of Wang. This result is applied to homological ideals and idempotents to produce new reduction techniques for testing the properties of syzygy-finite and injectives generation of finite dimensional algebras over a field.

Double Lie algebras of a nonzero weight

We introduce the notion of λ-double Lie algebra, which coincides with usual double Lie algebra when λ=0. We show that every λ-double Lie algebra for λ≠0 provides the structure of modified double Poisson algebra on the free associative algebra. In particular, it confirms the conjecture of S. Arthamonov (2017). We prove that there are no simple finite-dimensional λ-double Lie algebras.

Duality for Optimal Couplings in Free Probability

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on {mathbb {R}}^m are replaced by non-commutative laws of m-tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative L^2-Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X, Y) is a pair of optimally coupled m-tuples of non-commutative random variables in a tracial mathrm {W}^*-algebra mathcal {A}, then mathrm {W}^*((1 - t)X + tY) = mathrm {W}^*(X,Y) for all t in (0,1). Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m-tuples is not separable with respect to the Wasserstein distance for m > 1.

Solution of ill-posed problems with Chebfun

The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required.