Triangular Algebras Research Articles

Recollements of Singularity Categories and Monomorphism Categories

We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules.

Jordan σ-derivations of triangular algebras

We consider the problem of describing the form Jordan -derivations of a triangular algebra . The main result states that every Jordan -derivation of is of the form , where is a -derivation of and is a special mapping of . We search for sufficient conditions on a triangular algebra, such that . In particular, any Jordan -derivation of a nest algebra is a -derivation and any Jordan -derivation of an upper triangular matrix algebra , where is a commutative unital algebra, is a -derivation.

Hom-ideal structure of monoidal Hom-algebras in a category

In this paper, firstly we study the central invariant ZH(L)0 of generalized Hom-Lie algebras L, which are monoidal Hom-Lie algebras in the category [Formula: see text] of H-modules for a triangular Hopf algebra (H, R). If V is an H-Hom-Lie ideal of [L, L], under certain conditions we obtain V0 ⊆ ZH(L)0, where V0 is the monoidal Hom-subalgebra of H-invariant of V. Secondly, we describe the H-Hom-Lie ideal structure of monoidal Hom-algebras in a module category.

Zero triple product determined generalized matrix algebras

In this paper, we prove that the generalized matrix algebra G = \left[ A M N B \right] is a zero triple product (resp. zero Jordan triple product) determined if and only if A and B are zero triple products (resp. zero Jordan triple products) determined under certain conditions. Then the main results are applied to triangular algebras and full matrix algebras.

Gorenstein Projective Modules Over a Class of Generalized Matrix Algebras and their Applications

In this article, we introduce a class of generalized matrix algebras, in which each algebra is called a normally upper triangular gm algebra, and characterise Gorenstein projective modules over this class of algebras. Moreover, a sufficient condition of strongly Gorenstein projective modules over normally upper triangular gm algebras is given. The importance of normally upper triangular gm algebras for us is that it includes the so-called path algebras of quivers over algebras and generalized path algebras. Due to this, we characterize Gorenstein projective modules and strongly Gorenstein projective modules over path algebras of quivers over algebras and generalized path algebras as applications of the main results on normally upper triangular gm algebras. At last, we give an example to show how all indecomposable Gorenstein projective modules over a given algebra are constructed by the result on generalized path algebras.

On the Mahler measure of the Coxeter polynomial of an algebra

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformationϕA(T) as the automorphism of the Grothendieck group K0(A) induced by the Auslander–Reiten translation τ in the derived category Derb(modA) of the module category modA of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χA of ϕA is a product of cyclotomic polynomials, equivalently, if the Mahler measureM(χA)=1. In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T10+T9−T7−T6−T5−T4−T3+T+1 has Mahler measure μ0=1.176280..., and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M(χA)=1 or M(χB)≥μ0 for some convex subcategory B of A. We introduce interlaced tower of algebrasAm,…,An with m≤n−2 satisfyingχAs+1=(T+1)χAs−TχAs−1 for m+1≤s≤n−1. We prove that, if SpecϕAn⊂S1∪R+ and An is not of cyclotomic type, then M(χAm)<M(χAn). We construct several examples.

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Let be a generalized matrix algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and is the trace of . We describe the form of satisfying the condition for all . The question of when has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

A note on f-derivations of triangular algebras

Let \({\mathcal{A}}\) be a triangular algebra over a commutative ring R. We study f-derivations \({d:\mathcal{A} \rightarrow \mathcal{A}}\), i.e. linear maps, that act like derivations on a fixed multilinear polynomial f with coefficients from R. We show that every f-derivation d of \({\mathcal{A}}\), where the sum of coefficients of the polynomial f is not zero, is a derivation.

Centralizing traces and Lie triple isomorphisms on triangular algebras

Let be a triangular algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and that is a trace of . We describe the form of satisfying the condition for all . The question of when has the standard form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on to be almost standard. As applications, we characterize Lie triple isomorphisms of upper triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

GENERALIZED HOM-LIE STRUCTURE OF MONOIDAL HOM-ALGEBRAS IN A CATEGORY

In this paper, we study the structure of monoidal Hom-Lie algebras in the category Hℳ of H-modules for a triangular Hopf algebra (H, R) and in particular the H-Lie structure of a monoidal Hom-algebra in Hℳ by analogy with that of generalized Lie algebras.

Lie triple derivations of unital algebras with idempotents

Let be a unital algebra with a nontrivial idempotent over a unital commutative ring . We show that under suitable assumptions, every Lie triple derivation on is of the form , where is a derivation of , is a singular Jordan derivation of and is a linear mapping from to its centre that vanishes on . As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras.

Graded triangular algebras

The structure of graded triangular algebras T of arbitrary dimension are studied in this paper. This is motivated in part for the important role that triangular algebras play in the study of oriented graphs, upper triangular matrix algebras or nest algebras. It is shown that T decomposes as T = U + (∑i∈I Ti), where U is an R-submodule contained in the 0-homogeneous component and any Ti a well-described (graded) ideal satisfying TiTj = 0 if i≠j. Since any T is not simple asassociative algebra, the concept of quasi-simple triangular algebra is introduced as those T which are as near to simplicity as possible. Under mild conditions, the quasi-simplicity of T is characterized and it is proven that T is the direct sum of quasi-simple graded triangular algebras which are also ideals.

Nonlinear Jordan Triple Derivations of Triangular Algebras

In this paper, it is proved that every nonlinear Jordan triple derivation on triangular algebra is an additive derivation.

Commuting traces and Lie isomorphisms on generalized matrix algebras

Let G be a generalized matrix algebra over a commutative ring R, q: G × G ! G be an R-bilinear mapping and Tq: : G ! G be a trace of q. We describe the form of Tq satisfying the condition Tq(G)G = GTq(G) for all G 2 G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie isomorphism of G to be almost standard. As applications we characterize Lie isomorphisms of full matrix algebras, of triangular algebras and of certain uni- tal algebras with nontrivial idempotents. Some further research topics related to current work are proposed at the end of this article.

Generalized derivations on unital algebras determined by action on zero products

Let A be a unital algebra having a nontrivial idempotent and let M be a unitary A-bimodule. We consider linear maps F,G:A→M satisfying F(x)y+xG(y)=0 whenever x,y∈A are such that xy=0. For example, when A is zero product determined algebra (e.g. algebra generated by idempotents) F and G are generalized derivations F(x)=F(1)x+D(x) and G(x)=xG(1)+D(x) for all x∈A, where D:A→M is a derivation. If A is not generated by idempotents then there exist also nonstandard solutions for maps F and G. In the case of A being a triangular algebra under some condition on bimodule M the characterization of maps F and G is given. We also consider conditions on algebra A making it a zero product determined algebra.

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G n , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.

Zero product determined triangular algebras

Let 𝒜 be an algebra over a commutative unital ring 𝒞. We say that 𝒜 is zero product determined if for every 𝒞-module 𝒱 and every 𝒞-bilinear map φ: 𝒜 × 𝒜 → 𝒱 the following holds: if φ(A, B) = 0 whenever AB = 0, then there exists a 𝒞-linear map L such that φ(A, B) = L(AB) for all A, B ∈ 𝒜. If we replace, in this definition, the ordinary product by the Jordan (resp. Lie) product, then we say that 𝒜 is zero Jordan (resp. Lie) product determined. We show that the triangular algebra is zero (resp. Lie) product determined if and only if 𝒜 and ℬ are zero (resp. Lie) product determined, and under some technical restrictions, a same result is true for the Jordan product. The main result is then applied to generalized triangular matrix algebras and (block) upper triangular matrix algebras. In particular, we show that: (i) the block upper triangular matrix algebra (n ≥ 1) is a zero product determined algebra; (ii) if 𝒞 contains the element , then (n ≥ 1) is a zero Jordan product determined algebra and (iii) if 𝒜 is a commutative algebra, then (n ≥ 1) is a zero Lie product determined algebra.

Characterizations of additive (generalized) ξ-Lie (α, β) derivations on triangular algebras

Let 𝒯 be a triangular algebra over a real or complex field 𝔽 and α, β be the automorphisms of 𝒯. For a scalar ξ ∈ 𝔽, an additive mapping L: 𝒯 → 𝒯 is called a ξ-Lie (α, β) derivation if L([x, y]ξ) = L(x)α(y) + β (x)L(y) − ξL(y)α (x) − ξβ(y)L(x) for any x, y ∈ 𝒯. In this article, we characterized the additive ξ-Lie (α, β) derivation on triangular algebras. In addition, additive mappings ξ-Lie(α, β) derivable at some points are characterized.

The group of commutativity preserving maps on upper triangular matrices over a commutative ring

Let 𝒯 = 𝒯 n (R) be the associative algebra of all n × n upper triangular matrices over a unital commutative ring R with n > 2. A map σ on 𝒯 is called preserving commutativity in both directions if xy = yx ⇔ σ(x)σ(y) = σ(y)σ(x). For an invertible linear map σ on 𝒯, the following two conditions are shown to be equivalent: (a) σ preserves commutativity in both directions and (b) σ takes the form: σ(X) = cS −1[ϵX + (ϵ − 1)PX′P]S + f(X)I, ∀X ∈ 𝒯, where c ∈ R is invertible, ϵ ∈ R is idempotent, i.e. ϵ2 = ϵ, S ∈ 𝒯 is invertible, , X′ means the transpose of X and f is a linear function from 𝒯 to R such that 1 + f(I) is invertible. This result extends the main theorem of Marcoux and Sourour [L.W. Marcoux and A.R. Sourour, Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras, Linear Algebra Appl. 288 (1999), pp. 89–104] to an arbitrary commutative ring.

N-Derivations of triangular algebras

Let A be a triangular algebra. Let n⩾2 be an integer. A map φ:A×A×⋯×A→A is said to be a n-derivation if it is a derivation in each argument. In this paper we investigate n-derivations (n⩾3) for a certain class of triangular algebras. The main result is then applied to upper triangular matrix algebras and nest algebras.