Related papers: Duality in Derived Category $\mathcal O^\infty$

Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. We…

This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic…

We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…

The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary,…

Given a torsion pair $(\mathcal{T},\mathcal{F})$ in an abelian category $\mathcal{A}$ and its Happel-Reiten-Smal{\o} tilt $\mathcal{B}$, the equivalence of the realization functor $D^b({\mathcal B})\to D^b({\mathcal A})$ is determined by…

The authors define a Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra $\mathfrak{g}$ with respect to a triangular decomposition. This much needed approach unifies many important constructions in the existing literature in a…

We introduce a tensor category O_+ (resp. O_{-}) of certain modules of gl_{\infty} with non-negative (resp. non-positive) integral central charges with the usual tensor product. We also introduce a tensor category O_f consisting of certain…

We study a category of modules over $\mathfrak{gl}(\infty)$ analogous to category $\mathcal O$. We fix adequate Cartan, Borel and Levi-type subalgebras $\mathfrak h, \mathfrak b$ and $\mathfrak l$ with $\mathfrak l \cong…

For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we introduce a category ${\mathcal O}^P$ which is equipped with a forgetful functor to the parabolic category ${\mathcal…

This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of…

Let X be a smooth elliptic fibration over a smooth base B. Under mild assumptions, we establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an O^* gerbe over a genus one fibration which is a…

Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type…

We develop the theory of a category ${\mathscr C}_A$ which is a generalisation to non-restricted ${\mathfrak g}$-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted ${\mathfrak g}$-modules, where…

We introduce the notion of the $\infty$-category of (complete) derived $G$-graded modules over a $G$-graded ring $R$ for a torsion-free abelian group $G$, and we study its foundational properties. Moreover, we prove a categorical…

We study category $\mathcal{O}$ for Takiff Lie algebras $\mathfrak{g} \otimes \mathbb{C}[\epsilon]/(\epsilon^2)$ where $\mathfrak{g}$ is the Lie algebra of a reductive algebraic group over $\mathbb{C}$. We decompose this category as a…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category $\mathcal O$ of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component…

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific…

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to…