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Register a new userDerived equivalences for a class of PI algebras
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We provide a description of the tilting complexes of a PI algebra whose spectrum is canonical homeomorphic to the one of its center.
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We prove a version of Koszul duality and the induced derived equivalence for Adams connected $A_infty$-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernv{s}te{ui}n-Gelfand-Gelfand correspondence for Adams connected $A_infty$-algebras. We give various applications. For example, a connected graded algebra $A$ is Artin-Schelter regular if and only if its Ext-algebra $Ext^ast_A(k,k)$ is Frobenius. This generalizes a result of Smith in the Koszul case. If $A$ is Koszul and if both $A$ and its Koszul dual $A^!$ are noetherian satisfying a polynomial identity, then $A$ is Gorenstein if and only if $A^!$ is. The last statement implies that a certain Calabi-Yau property is preserved under Koszul duality.
Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this paper we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras.
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 show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.
We present a proof of Kemers representability theorem for affine PI algebras over a field of characteristic zero.
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