Generalized Matrix Algebras Research Articles

On nonlinear Lie centralizers of generalized matrix algebras

ABSTRACT The aim of the paper is to give a description of nonlinear Lie centralizers for a certain class of generalized matrix algebras. The result is then applied to some full matrix algebras and triangular algebras.

σ-derivations on generalized matrix algebras

Abstract Let 𝒭 be a commutative ring with unity, 𝒜, 𝒝 be 𝒭-algebras, 𝒨 be (𝒜, 𝒝)-bimodule and 𝒩 be (𝒝, 𝒜)-bimodule. The 𝒭-algebra 𝒢 = 𝒢(𝒜, 𝒨, 𝒩, 𝒝) is a generalized matrix algebra defined by the Morita context (𝒜, 𝒝, 𝒨, 𝒩, ξ𝒨𝒩, Ω𝒩𝒨). In this article, we study Jordan σ-derivations on generalized matrix algebras.

Characterizations of Lie triple derivations on generalized matrix algebras

Let be a commutative ring with unity and be a generalized matrix algebra. In this article, we give the structure of Lie triple derivation on a generalized matrix algebra and prove that under certain appropriate assumptions on is proper, i.e., where δ is a derivation on and χ is a mapping from into its center which annihilates all second commutators in i.e., for all

On local Lie derivations of generalized matrix algebras

Let $${\mathcal {G}}$$ be a generalized matrix algebra. We prove that, under certain conditions, every local Lie derivation $$\delta $$ of $${\mathcal {G}}$$ can be written in the form $$\delta =d+h$$, where d is a derivation of $${\mathcal {G}}$$ and h is a linear map from $${\mathcal {G}}$$ into $${\mathcal {Z}}({\mathcal {G}})$$ vanishing on each commutator. The result is then applied to some full matrix algebras and triangular algebras.

Multiplicative generalized Lie triple derivations on generalized matrix algebras

Let be a commutative ring with unity, be -algebras, be ()-bimodule and be ()-bimodule. The -algebra is a generalized matrix algebra defined by the Morita context . In this article, we study multiplicative generalized Lie triple derivation on generalized matrix algebras and prove that every multiplicative generalized Lie triple derivation on a generalized matrix algebra has the standard form.

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We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the n-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.

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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.

Characterizations of (m,n)-Jordan Derivations and (m,n)-Jordan Derivable Mappings on Some Algebras

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and $$\mathcal{U}=\begin{bmatrix}\mathcal{A} & \mathcal{M} \\\mathcal{N} & \mathcal{B} \end{bmatrix}$$ is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.

K-commuting mappings of generalized matrix algebras

In this paper we will study k-commuting mappings of generalized matrix algebras. The general form of arbitrary k-commuting mapping of a generalized matrix algebra is determined. It is shown that under mild assumptions, every k-commuting mapping of a generalized matrix algebra takes a certain form which is said to be proper. A number of applications related to k-commuting mappings are presented. These results extend the existing works of Cheung (J Lond Math Soc 63:117–127, 2001), Du and Wang (Linear Algebra Appl 436:1367–1375 2012) to the case of generalized matrix algebras.

More on Lie derivations of a generalized matrix algebra

More on Lie derivations of a generalized matrix algebra

The First Hochschild Cohomology of Square Algebras With it's Stability

In this paper, we study on a special case of generalized matrix algebra that we call it square algebra. According to that Hochschild cohomology play a significant role in Geometry for example in orbifolds, we study the first Hochschild cohomology of the square algebra the vanishing of its.

Superalgebraic representation of Dirac matrices

We consider a Clifford extension of the Grassmann algebra in which operators are constructed from products of Grassmann variables and derivatives with respect to them. We show that this algebra contains a subalgebra isomorphic to a matrix algebra and that it additionally contains operators of a generalized matrix algebra that mix states with different numbers of Grassmann variables. We show that these operators are extensions of spin-tensors to the case of superspace. We construct a representation of Dirac matrices in the form of operators of a generalized matrix algebra.

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.

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.

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.

ADDITIVITY OF JORDAN TRIPLE PRODUCT HOMOMORPHISMS ON GENERALIZED MATRIX ALGEBRAS

In this article, it is proved that under some conditions every bijective Jordan triple product homomorphism from generalized matrix algebras onto rings is additive. As a corollary, we obtain that every bijective Jordan triple product homomorphism from <TEX>$M_n(\mathcal{A})$</TEX> (<TEX>$\mathcal{A}$</TEX> is not necessarily a prime algebra) onto an arbitrary ring <TEX>$\mathcal{R}^{\prime}$</TEX> is additive.

Triangular matrix coalgebras and applications

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that is a coalgebra such that the rational part of every left finitely generated -module splits off if and only if is an upper triangular matrix coalgebra, for a serial coalgebra whose Ext-quiver is a finite union of cycles, a finite dimensional coalgebra and a finite dimensional bicomodule .

Biderivations of generalized matrix algebras

The aim of the paper is to give a description of biderivations for a certain class of generalized matrix algebras. We prove that under certain conditions a biderivation of a generalized matrix algebra is a sum of an extremal and an inner biderivation. We also consider the question when a biderivation of a generalized matrix algebra is an inner biderivation. As a consequence we show that every biderivation of a full matrix algebra over a unital algebra is inner.

Jordan derivations and antiderivations of generalized matrix algebras

Let $\mathcal{G}=[A & M N & B]$ be a generalized matrix algebra defined by the Morita context $(A, B,_AM_B,_BN_A, \Phi_{MN}, \Psi_{NM})$. In this article we mainly study the question of whether there exist proper Jordan derivations for the generalized matrix algebra $\mathcal{G}$. It is shown that if one of the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ is nondegenerate, then every antiderivation of $\mathcal{G}$ is zero. Furthermore, if the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ are both zero, then every Jordan derivation of $\mathcal{G}$ is the sum of a derivation and an antiderivation. Several constructive examples and counterexamples are presented.