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In this paper, we study the BGG category $mathcal{O}$ for the quantum Schr{o}dinger algebra $U_q(mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $dot z eq 0$, using the module $B_{dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $mathcal{O}[dot z]$ consisting of modules with the central charge $dot z$ and the BGG category $mathcal{O}^{(mathfrak{sl}_2)}$ for the quantum group $U_q(mathfrak{sl}_2)$. In the case that $dot z=0$, we study the subcategory $mathcal{A}$ consisting of finite dimensional $U_q(mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in cite{DLMZ, Mak}, we directly construct an equivalent functor from $mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(mathfrak{s})$-modules is wild.
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The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) infinitesimal Hecke algebras, in addition to semisimple Lie algebras and their quantum groups. We then present numerous examples, two families of which are studied in detail. These are quantum groups defined using not necessarily the root or weight lattices (for these, we study the center and central characters), and infinitesimal Hecke algebras.
This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $Gamma$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A rtimes Gamma$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the Conditions (S); the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup - and which yield information about O.
For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztigs canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.
Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category ${mathcal{O}}$. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category ${mathcal{O}}$. In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in ${mathcal{S}}$-subcategories in ${mathcal{O}}$.
We determine the Verma multiplicities and the characters of projective modules for atypical blocks in the BGG Category O for the general linear Lie superalgebras $frak{gl}(2|2)$ and $frak{gl}(3|1)$. We then explicitly determine the composition factor multiplcities of Verma modules in the atypicality 2 block of $frak{gl}(2|2)$.
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