Periodic trivial extension algebras and fractionally Calabi-Yau algebras

Recently the relations between tiltingtheory and trivialextension algebras are deeply studied. Let A and B be basic connected artin algebras over a commutative artin ring C. In [6] Tachikawa and Wakamatsu showed that the existence of stably equivalence between categories over the trivialextension algebras T(A)=A kDA and T(B)=Bt<DB under the assumption that there is a tiltingmodule TA with B=End(TA). In case C is a field,Hughes and Waschblisch proved that if T(B) is representation-finiteof Cartan class A, then there exists a tiltedalgebra A of Dynkin type A such that T(B)^T(A) [4]. Assem, Happel and Roldan showed that, for an algebra B over an algebraically closed field, T(B) is representation-finiteiff B is an iterated tiltedalgebra of Dynkin type [1]. However in case T(B) is not of finiterepresentation type the condition T{B) = T{A) with A hereditary does not forces B to be an iterated tiltedalgebra. Let's consider the covering A of T(A) [4]. The author proved that the condition A^B implies T(A) = T(B) and that the converse holds if T{A) is representation-finite[5]. In this paper, we prove that the condition B = A with A hereditary implies that B is an iterated algebra obtained from A. It is to be noted that in case A is not necessary representation-finite. Moreover, the proof of our theorem shows that such an algebra B is obtained by at most 3m times processes tilting from A, where m is the number of non-isomorphic primitive idempotents of A.

For a bound quiver algebra satisfying the condition that the every oriented cycles in the quiver are vanished in the algebra, Fernández and Platzeck determined the bound quiver algebra which is isomorphic to the trivial extension algebra. In this paper, we consider a Hochschild extension algebra which is a generalization of a trivial extension algebra. The purpose of this paper is to determine the bound quiver algebras which are isomorphic to Hochschild extension algebras of some finite dimensional self-injective Nakayama algebras.

In this paper, we investigate the problem of describing the form of Lie generalized derivations on trivial extension algebras. We show, under some conditions, that every Lie generalized derivation on a trivial extension algebra is an sum of a generalized derivation and a center valued map which vanishes on all commutators. As an application we characterize Lie generalized derivation on a triangular algebra.

In this paper, we investigate, in detail, derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence, we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite.

Graded self-injective algebras “are” trivial extensions

Let A be a Poisson algebra and M be a left Poisson module. We construct a Poisson structure on the trivial extension algebra $$A\ltimes M$$ . We investigate, in detail, Poisson derivations and Hamiltonian derivations on $$A\ltimes M$$ . As a consequence, we characterize the first Poisson cohomology group of $$A\ltimes M$$ in terms of the ones of A and M. In the case of that A is finite dimensional and $$M=A^*$$ , we show that $${\text {HP}}^1(A)$$ is a summand of $${\text {HP}}^1(A\ltimes M)$$ . These are generalization of the results on the derivation and the first Hochschild cohomology group of trivial extension algebras in Assem et al. (J Pure Appl Algebra 220:2471–2499, 2016), Bennis and Fahid (Medieter J Math 14:150, https://doi.org/10.1007/s00009-017-0949-z , 2017), Cibils et al. (Glasg Math J 45:21–40, 2003) to Poisson framework.

In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.

Homological dimension formulas for trivial extension algebras

The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

Jordan Derivations on Trivial Extension Algebras

For a self-orthogonal module T, the relation between the quotient triangulated category D b(A)/K b(addT) and the stable category of the Frobenius category of T-Cohen-Macaulay modules is investigated. In particular, for a Gorenstein algebra, we get a relative version of the description of the singularity category due to Happel. Also, the derived category of a Gorenstein algebra is explicitly given, inside the stable category of the graded module category of the corresponding trivial extension algebra, via Happel’s functor \(F: D^b(A) \longrightarrow T(A)^{\mathbb{Z}}\mbox{-}\underline{\rm mod}\).

&lt;abstract&gt;&lt;p&gt;In this paper, we investigated the problem of describing the form of higher Jordan triple derivations on trivial extension algebras. We show that every higher Jordan triple derivation on a $ 2 $-torsion free $ * $-type trivial extension algebra is a sum of a higher derivation and a higher anti-derivation. As for its applications, higher Jordan triple derivations on triangular algebras are characterized.&lt;/p&gt;&lt;/abstract&gt;

Tilting functors and stable equivalences for selfinjective algebras

Quadratic forms and iterated tilted algebras