Trivial Extension Algebras Research Articles

Hochschild homology dimension and a class of Hochschild extension algebras of truncated quiver algebras

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite.

Nonlinear Jordan triple derivable mapping on $ * $-type trivial extension algebras

<abstract><p>The aim of the paper was to give a description of nonlinear Jordan triple derivable mappings on trivial extension algebras. We proved that every nonlinear Jordan triple derivable mapping on a $ 2 $-torsion free $ * $-type trivial extension algebra is a sum of an additive derivation and an additive antiderivation. As an application, nonlinear Jordan triple derivable mappings on triangular algebras were characterized.</p></abstract>

Higher Jordan triple derivations on $ * $-type trivial extension algebras

<abstract><p>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.</p></abstract>

Characterizations of Lie n -Centralizers on Certain Trivial Extension Algebras

In this paper, we describe the structure of Lie n -centralizers of a trivial extension algebra. We then present some conditions under which a Lie n -centralizer on a trivial extension algebra is proper. As an application, we consider Lie n -centralizers on a triangular algebra.

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n $n$ -dimensional affine hypersurface of degree n + 2 $n+2$ and the double cover of the n $n$ -dimensional affine space branched along a degree 2 n + 2 $2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.

Higher preprojective algebras, Koszul algebras, and superpotentials

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is$d$-hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a$d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras A and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where A is a trivial extension algebra A=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. In [17] we gave a necessary and sufficient condition that A is IG in terms of Λ and C by using derived tensor products and derived Homs. For simplicity, we assume that Λ is of finite global dimension in the sequel. In this paper, we show that the condition that A is IG, has a triangulated categorical interpretation. We prove that if A is IG, then the graded stable category CM_ZA of CM-modules is realized as an admissible subcategory of the derived category Db(modΛ). As a corollary, we deduce that the Grothendieck group K0(CM_ZA) is free of finite rank.We give several applications. Among other things, for a path algebra Λ=kQ of an A2 or A3 quiver Q, we give a complete list of Λ-Λ-bimodule C such that Λ⊕C is IG (resp. of finite global dimension) by using the triangulated categorical interpretation mentioned above.

Jordan centralizer maps on trivial extension algebras

AbstractThe 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.

Homological dimension formulas for trivial extension algebras

This paper is the first one of a series of papers in which we study a finitely graded Iwanaga-Gorenstein (IG) algebras A=⨁i=0ℓAi and their representation theory. Every finitely graded algebra is graded Morita equivalent to a trivial extension algebra A=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. Hence, in principle, representation theoretic study of finitely graded algebras is reduced to that of trivial extension algebras. The main result of this paper is a necessary and sufficient condition that a trivial extension algebra A=Λ⊕C to be an IG-algebra. We also give a description of the kernel Ker ϖ of a canonical functor ϖ:Db(modΛ)→SingZA. These results play important roles in the subsequent works.To this end, we study complexes of graded projective (injective) A-modules and establish formulas for the projective (injective) dimension of a graded A-modules in terms of Λ and C. As a byproducts, we obtain a modern expression of the classical formula for the global dimension of a trivial extension algebra by Palmer-Roos and Löfwall that is written in complicated classical derived functors.

On generalized Lie derivations

In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by Benkovic. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras.

Poisson Derivations and the First Poisson Cohomology Group on Trivial Extension Algebras

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.

Lie generalized derivations on trivial extension 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.

On presentations of Hochschild extension algebras for a class of self-injective Nakayama algebras

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.

Lie Derivations on Trivial Extension Algebras

Abstract 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.

Trivial extension of Koszul algebras

Let Λ be a Koszul algebra, and let M be a graded Λ-bimodule. We prove that the trivial extension algebra of Λ by M is also a Koszul algebra whenever M is Koszul as a left Λ-module. Applications and examples are also provided.

Derivations and the First Cohomology Group of Trivial Extension Algebras

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.

Hochschild homology and trivial extension algebras

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

Categorification of a linear algebra identity and factorization of Serre functors

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.

Singular Equivalences of Trivial Extensions

We prove that a certain pair of bimodules over two artin algebras gives rise to a triangle equivalence between the singularity categories of the two corresponding trivial extension algebras. Some consequences and an example are given.