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Register a new userOn the derived category of quasi-hereditary algebras with two simple modules
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Authors
Yury Volkov
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We describe the derived Picard groups and two-term silting complexes for quasi-hereditary algebras with two simple modules. We also describe by quivers with relations all algebras derived equivalent to a quasi-hereditary algebra with two simple modules.
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We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs $(A,M)$, where $A$ is the $s$-Veronese ring and $M$ is the $(s,1)$-Veronese bimodule of the $s$-homogeneous dual algebra. For each such a pair we give an intrinsic characterization of algebras corresponding to it. Due to results of our previous work many pairs determine the algebra uniquely up to isomorphism. Using our partial classification, we show that, to check the $s$-Koszulity of a connected $s$-homogeneous algebras with two relations, it is enough to verify an equality for Hilbert series or to check the exactness of the generalized Koszul complex in the second term. For each pair $(A,M)$ not belonging to one specific series of pairs, we check if there exists an $s$-Koszulity algebra corresponding to it. Thus, we describe a class of possible ${rm Ext}$-algebras of $s$-Koszul connected algebras with two relations and realize all of them except a finite number of specific algebras as ${rm Ext}$-algebras. Another result that follows from our classification is that an $s$-homogeneous algebra with two dimensional $s$-th component cannot be $s$-Koszul for $s>2$.
Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $mathcal{S}$ be a simple-minded system of the stable module category $stmod{A}$. We show that the intersection $mathcal{S}capmathcal{C}$ is of size strictly less than $n$, and consists only of modules with quasi-length strictly less than $n$. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of $A$ cannot be in any simple-minded system.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
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