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Register a new userDifferential graded algebras over some reductive group
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Authors
Jin Cao
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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.
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A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $mathbb{L}$ of $mathbb{F}$: (1) finite-dimensional central division algebras over $mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups over $mathbb{L}$, up to graded-isomorphism; (3) $mathbb{F}$-forms of certain graded matrix algebras with coefficients in $Deltaotimes_{mathbb{L}}mathcal{C}$ where $Delta$ is as in (1) and $mathcal{C}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.
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.
Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian subgroup $H$, a positive integer $d$ and an element of $Hom(M(H), k^times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main theorem is the converse: Given an extension $1rightarrow Hrightarrow Grightarrow G/Hrightarrow 1$, where $H$ is abelian, a positive integer $d$, and an element of $Hom(M(H), k^times)^G$, there is a division algebra with center containing $k$ that realizes these data. We apply this result to classify the $G$-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field.
We consider finite-dimensional irreducible transitive graded Lie algebras $L = sum_{i=-q}^rL_i$ over algebraically closed fields of characteristic three. We assume that the null component $L_0$ is classical and reductive. The adjoint representation of $L$ on itself induces a representation of the commutator subalgebra $L_0$ of the null component on the minus-one component $L_{-1}.$ We show that if the depth $q$ of $L$ is greater than one, then this representation must be restricted.
Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A#H$ is a dg algebra. For any dg $A#H$-module $M$, there is a quasi-isomorphism of dg algebras: $mathrm{RHom}_A(M,M)#Hlongrightarrow mathrm{RHom}_{A#H}(Mot H,Mot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras
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