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Register a new userA differential graded approach to the silting theorem
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A silting theorem was established by Buan and Zhou as a generalisation of the classical tilting theorem of Brenner and Butler. In this paper, we give an alternative proof of the theorem by using differential graded algebras.
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The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of g and whose weights are bounded by some dominant weight of g. We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl_n(A). Our results unify previous work of Chari and her collaborators on map algebras and of Seligman on isotropic Lie algebras.
In cite{CK2005} and cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2--Calabi--Yau (Auslander--Reiten formula) and high order associativity.
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Kellers approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas cite{Dugas2015} in some special case.
We introduce pre-silting and silting subcategories in extriangulated categories and generalize the silting theory in triangulated categories. We prove that the silting reduction $mathcal B/({rm thick}mathcal W)$ of an extriangulated category $mathcal B$ with respect to a pre-silting subcategory $mathcal W$ can be realized as a certain subfactor category of $mathcal B$. This generalizes the result by Iyama-Yang. In particular, for a Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang by this reduction.
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$ and $widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
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