We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.
In this paper, we first introduce $mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $xymatrix@C=80pt{mathcal {G}(mathcal {F})cap mathcal {A}_C(R) ar@<0.5ex>[r]^{Cotimes_R-} & mathcal {G}(mathcal {W}_F) ar@<0.5ex>[l]^{textrm{Hom}_R(C,-)}} $ where $mathcal {G}(mathcal {F})$, $mathcal {A}_C(R) $ and $mathcal {G}(mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $mathcal {W}_F$-Gorenstein if there exists an exact sequence $mathbb{G}_bullet=indent ...longrightarrow G_1longrightarrow G_0longrightarrow G^0longrightarrow G^1longrightarrow...$ in $mathcal {G}(mathcal {W}_F)$ such that $M cong$ $im(G_0rightarrow G^0) $ and that $mathbb{G}_bullet$ is Hom$_R(mathcal {G}(mathcal {W}_F),-)$ and $mathcal {G}(mathcal {W}_F)^+otimes_R-$ exact. We show that two notions of the two-degree $mathcal {W}_F$-Gorenstein and the $mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
This paper shows how to obtain the key concepts and notations of Garside theory by using the Composition--Diamond lemma. We also show that in some cases the greedy normal form is exactly a Grobner--Shirshov normal form and a family of a left-cancellative category is a Garside family, if and only if a suitable set of reductions is confluent up to some congruence on words.
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and double-free left modules (free modules over a free algebra). We give the Chibrikovs Composition-Diamond lemma for modules and show that Kang-Lees Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.
In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.