Triangular Matrix Algebras Research Articles

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

From subcategories to the entire module categories

Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

The image of polynomials on 2 × 2 upper triangular matrix algebras

The goal of the paper is to give a complete description of the image of polynomials with zero constant term on 2×2 upper triangular matrix algebras over an algebraically closed field.

Involutive triangular matrix algebras

In this paper we provide new examples of Banach $ \ast $-subalgebras of the matrix algebra $M_n(\mathscr{A}) $ over a commutative unital $C^*$-algebra $\mathscr{A}$. For any involutive algebra, we define two involutions on the triangular matrix extensions. We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal.*******************************************************************************

Lie σ-derivations of triangular algebras

ABSTRACT Let be a triangular algebra and σ be an automorphism of . We consider the problem of describing the form of Lie σ-derivations of . In particular, we give sufficient conditions that every Lie σ-derivation d of is the sum , where Δ is a σ-derivation of and γ is a linear mapping from to its σ-centre that vanishes on . As an application, Lie σ-derivations of (block) upper triangular matrix algebras and nest algebras are determined.

On deformations of Gorenstein-projective modules over Nakayama and triangular matrix algebras

Let k be a fixed field of arbitrary characteristic and let Λ be a basic connected Nakayama k -algebra without simple projective modules. In this article we prove that if V is an indecomposable finitely generated Gorenstein-projective left Λ-module, then the versal deformation ring R ( Λ , V ) (in the sense of F. M. Bleher and the author) is universal and stable after taking syzygies . We also prove the following result. Let Σ = ( Λ B 0 Γ ) be a triangular matrix finite dimensional Gorenstein k -algebra with B projective as a left Λ-module and with Γ of finite global dimension. If ( V W ) f is a finitely generated Gorenstein-projective left Σ-module with End _ Σ ( ( V W ) f ) = k , then V is also a finitely generated Gorenstein-projective left Λ-module with End _ Λ ( V ) = k , and the versal deformation rings R ( Σ , ( V W ) f ) and R ( Λ , V ) are both universal and isomorphic.

The Images of Completely Homogeneous Polynomials on 2 × 2 Upper Triangular Matrix Algebras

The purpose of this paper is to initiate the study of the images of non-multilinear polynomials on upper triangular matrix algebras. We shall give a complete description of the images of completely homogenous polynomials on 2 × 2 upper triangular matrix algebras. As a consequence, we show that the image of some special completely homogenous polynomials on 2 × 2 upper triangular matrix algebras are not vector spaces.

Eventually homological isomorphisms in recollements of derived categories

Eventually homological isomorphisms are introduced between derived categories of algebras, and these lead to a special recollement situation, that is, (DB,DA,DC) with the functor j⁎:DA→DC being an eventually homological isomorphism. This recollement turns out to be a very good context to compare the algebras A and C with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.

Automorphisms of a certain subalgebra of the upper triangular matrix algebra

For a commutative ring $R$ with unity, the $R$-algebra of strictly upper triangular $n\times n$ matrices over $R$ is denoted by $N_{n}\left( R\right) $, where $n$ is a positive integer greater than $1$. For the identity matrix $I$, $\alpha \in R$, $A \in N_n(R)$, the set of all elements $\alpha I+A$ is defined as the scalar upper triangular matrix algebra $ST_n(R)$ which is a subalgebra of the upper triangular matrices $T_n(R) .$ In this paper, we investigate the $R$-algebra automorphisms of $ST_{n}\left( R\right) .$ We extend the automorphisms of $N_{n}\left( R\right) $ to $ST_{n}\left( R\right)$ and classify all the automorphisms of $ST_{n}\left( R\right) .$ We generalize the results of Cao and Wang and prove that not all automorphisms of $ST_{n}\left( R\right) $ can be extended to the automorphisms of $T_{n}(R).$

Graded polynomial identities for the upper triangular matrix algebra over a finite field

Let K be a finite field and let UTn(K) be the algebra of n×n upper triangular matrices over K. In this paper we describe the set of all G-graded polynomial identities of UTn(K), where G is any group. Moreover, we describe a linear basis for the corresponding relatively free graded algebra.

On the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups

Let F be an algebraically closed field of characteristic zero and G be an arbitrary finite cyclic group. In this paper, given an m-tuple (A1,…,Am) of finite dimensional G-simple algebras, we focus on the study of the factorability of the TG-ideals IdG((UT(A1,…,Am),α˜)) of the G-graded upper block triangular matrix algebras UT(A1,…,Am) endowed with elementary G-gradings induced by some maps α˜.When G is a cyclic p-group we prove that the factorability of the ideal IdG((UT(A1,…,Am),α˜) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1,…,Am, as well to the existence of a unique isomorphism class of α˜-admissible elementary G-gradings for UT(A1,…,Am). Moreover, we present some necessary and sufficient conditions to the factorability of IdG((UT(A1,A2),α˜)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A1 and A2.

Characterizing Jordan n-Derivations of Unital Rings Containing Idempotents

Assume that $${\mathcal {R}}$$ is a unital ring containing a nontrivial idempotent. By introducing the concept of Jordan n-derivations (n is any positive integer), it is shown that, under certain conditions, every multiplicative (without any linearity and additivity) Jordan n-derivation $$\delta $$ on $${\mathcal {R}}$$ is additive; and furthermore, $$\delta $$ is a Jordan derivation if the characteristic of $${\mathcal {R}}$$ is not $$n-1$$ and $$\delta $$ is a generalized Jordan derivation if the characteristic of $${\mathcal {R}}$$ is $$n-1$$ $$(n>3)$$ . Based on these results, it turns out that a map on $${\mathcal {R}}$$ is a multiplicative Jordan n-derivation if and only if it is an additive Jordan derivation. As applications, multiplicative Jordan n-derivations on triangular rings, prime rings, matrix algebras, nest algebras and von Neumann algebras are, respectively, completely characterized, which generalize some known related results.

Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products

Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products

Left Ideal Preserving Maps on Triangular Algebras

Let A be a unital algebra over a commutative unital ring R. We say that A is a SLIP algebra if every R-linear map on A that leaves invariant every left ideal of A is a left multiplier. In this paper we study whether a triangular algebra Tri(A,M,B) is a SLIP algebra and give some necessary or sufficient conditions for a triangular algebra be a SLIP algebra, and various examples are given which illustrate limitations on extending some of the theory developed. Then our results are applied to generalized triangular matrix algebras and block upper triangular algebras. Also, some SLIP algebras other that triangular algebras are provided.

Generalized notions of amenability for a class of matrix algebras

We investigate the amenability and its related homological notions for a class of $I\times I$-upper triangular matrix algebra, say ${\rm UP}(I,A)$, where $A$ is a Banach algebra equipped with a nonzero character. We show that ${\rm UP}(I,A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of ${\rm UP}(I,A)$.

A characterization of minimal varieties of [formula omitted]-graded PI algebras

Let F be a field of characteristic zero and p a prime. In the present paper it is proved that a variety of Zp-graded associative PI F-algebras of finite basic rank is minimal of fixed Zp-exponent d if, and only if, it is generated by an upper block triangular matrix algebra UTZp(A1,…,Am) equipped with a suitable elementary Zp-grading, whose diagonal blocks are isomorphic to Zp-graded simple algebras A1,…,Am satisfying dimF⁡(A1⊕⋯⊕Am)=d.

Generalized Lie n-derivations of triangular algebras

Let be a triangular algebra. We show that under suitable assumptions every generalized Lie n-derivation associated with a linear map is of the form where and Δ is a Lie n-derivation of We solve this problem using commuting and centralizing maps. We also prove that under certain mild conditions any centralizing map on a triangular algebra is commuting. As an application, we give a description of generalized Lie n-derivations on classical examples of triangular algebras: upper triangular matrix algebras and nest algebras.

English

Let ℛ be a commutative ring, $$\mathcal{G}$$ be a generalized matrix algebra over ℛ with weakly loyal bimodule and $$\mathcal{G}$$ be the center of $$\mathcal{G}$$ . Suppose that $$\mathfrak{q} : \mathcal{G} \times \mathcal{G} \rightarrow \mathcal{G}$$ is an ℛ-bilinear mapping and that $$\mathfrak{T}_{\mathfrak{q}} : \mathcal{G} \rightarrow \mathcal{G}$$ is a trace of $$\mathfrak{q}$$ . The aim of this article is to describe the form of $$\mathfrak{T}_{\mathfrak{q}}$$ satisfying the centralizing condition $$\left[\mathfrak{T}_{\mathfrak{q}}(x), x\right] \in \mathcal{Z}(\mathcal{G})$$ (and commuting condition $$\left[\mathfrak{T}_{\mathfrak{q}}(x), x\right]=0 )$$ for all $$x \in \mathcal{G}$$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $$\mathfrak{T}_{\mathfrak{q}}$$ has the so-called proper form from a new perspective. Using the aforementioned trace function, we establish sufficient conditions for each Lie-type isomorphism of $$\mathcal{G}$$ to be almost standard. As applications, centralizing (commuting) traces of bilinear mappings and Lie-type isomorphisms on full matrix algebras and those on upper triangular matrix algebras are totally determined.

The images of multilinear polynomials on 2 × 2 upper triangular matrix algebras

ABSTRACTThe purpose of this paper is to describe the images of non-commutative multilinear polynomials on upper triangular matrix algebras.

Involutive operator algebras

Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the {\em complex symmetric} operators studied by Garcia, Putinar, Wogen and others. We investigate the general theory of involutive operator algebras, and give many applications.