Local Jordan Derivations and Local Jordan Automorphisms of Upper Triangular Matrix Algebras

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.

Representations of Frobenius-type Triangular Matrix Algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

Decomposition of Jordan automorphism of two-order upper triangular matrix algebra over certain semilocal rings

Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.

Gluing support τ-tilting modules via symmetric ladders of height 2

Gluing techniques with respect to a recollement have long been studied. Recently, ladders of recollements of abelian categories were introduced as important tools. In this paper, we present explicit constructions of gluing support τ-tilting modules via symmetric ladders of height two. Moreover, we apply the result to triangular matrix algebras to give a detailed version of the known Jasso's reduction and study maximal green sequences.

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.