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Register a new userKoszul differential graded algebras and BGG correspondence
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The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra $A$ is AS-regular, the Ext-algebra $E$ of $A$ is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left $E$-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG $A$-modules consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can derived from the DG version.
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We study the general properties of commutative differential graded algebras in the category of representations over a reductive algebraic group with an injective central cocharacter. Besides describing the derived category of differential graded modules over such an algebra, we also provide a criterion for the existence of a t-structure on the derived category together with a characterization of the coordinate ring of the Tannakian fundamental group of its heart.
In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($kgeq 5$) of these algebras are split.
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