Related papers: Borel and shifted category O

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by…

We generalize the notion of a Kac-Moody Lie algebra to the setting of Deligne Categories. Then we derive the Kac-Weyl formula for the category $\mathcal{O}$ integrable representations for such an algebra. This paper generalizes results of…

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 $G$ 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…

We consider a general parabolic category $\mathcal{O}_{S}$ over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on…

For an arbitrary affine Lie algebra we study an analog of the category O for the natural Borel subalgebra and zero central charge. We show that such category is semisimple having the reduced imaginary Verma modules as its simple objects.…

We give the first positive formulas for the weights of every simple highest weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we also express the weights of $L(\lambda)$ as an…

Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that…

We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an…

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type…

We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements…

We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by…

We study the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor, from O to the category of H-modules, where H is…

We determine the exact complexity of classifying compact metric spaces up to homeomorphism. More precisely, the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish…

We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter not a half integer), providing thus character formulas for simple modules. We give some generalization to…

We show that the category of Lie triple systems is equivalent to the category of Lie algebras graded by Z/(2Z) such that the odd component generates the algbera and the second graded cohomology group coefficients in any trivial module is…

We study the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme G((t))/I, where I is the Iwahori subgroup. We prove a…

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 formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma…

We classify the simple modules for the rational Cherednik algebra that are irreducible when restricted to W, in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in…