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Register a new userThe Hall algebra of a spherical object
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We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.
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Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid, as introduced by M. Reineke, and the generic composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid algebra of the composition monoid. Moreover, if Q is an acyclic, extended Dynkin quiver, we show that there exists an epimorphism from the composition algebra at q=0 to the monoid algebra of the composition monoid, and we describe its non-trivial kernel. Our main tool is a geometric version of BGP reflection functors on quiver Grassmannians and quiver flags, that is varieties consisting of filtrations of a fixed representation by subrepresentations of fixed dimension vectors. These functors enable us to calculate various structure constants of the composition algebra. Moreover, we investigate geometric properties of quiver flags and quiver Grassmannians, and show that under certain conditions, quiver flags are irreducible and smooth. If, in addition, we have a counting polynomial, these properties imply the positivity of the Euler characteristic of the quiver flag.
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.
We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shojis modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy elements for Y are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.
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.
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
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