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Register a new user$mathbb Z Q$ type constructions in higher representation theory
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Jin Yun Guo
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Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $mathbb Z Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. We introduce the $n$-translation quiver $mathbb Z|_{n-1} Q$ as a generalization of the $mathbb Z Q$ construction in our recent study on $n$-translation algebras. In this paper, we introduce $n$-slice algebra and show that for certain $n$-slice algebra $Gamma$, % of global dimension $n$, the quiver $mathbb Z|_{n-1} Q$ can be used to describe the $tau_n$-closure of $DGamma$ and $tau_n^{-1}$-closure of $Gamma$ in its module category and the $ u_n$-closure of $Gamma$ in the derived category.
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The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$ and $widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the algebra from 6 to 4 (or even 3) without loss of representation theoretic generality. We then find some non-semisimple specialisations by calculating Gram determinants for certain cell modules (or standard modules) using the good parametrisation defined. We finish by considering some quotients of specialisations of the symplectic blob algebra which are isomorphic to Temperley--Lieb algebras of type $A$.
For a commutative algebra $A$ over $mathbb{C}$,denote $mathfrak{g}=text{Der}(A)$. A module over the smash product $A# U(mathfrak{g})$ is called a jet $mathfrak{g}$-module, where $U(mathfrak{g})$ is the universal enveloping algebra of $mathfrak{g}$.In the present paper, we study jet modules in the case of $A=mathbb{C}[t_1^{pm 1},t_2]$.We show that $A#U(mathfrak{g})congmathcal{D}otimes U(L)$, where $mathcal{D}$ is the Weyl algebra $mathbb{C}[t_1^{pm 1},t_2, frac{partial}{partial t_1},frac{partial}{partial t_2}]$, and $L$ is a Lie subalgebra of $A# U(mathfrak{g})$ called the jet Lie algebra corresponding to $mathfrak{g}$.Using a Lie algebra isomorphism $theta:L rightarrow mathfrak{m}_{1,0}Delta$, where $mathfrak{m}_{1,0}Delta$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $mathfrak{g}$ with uniformly bounded weight spaces.
For a complex finite-dimensional simple Lie algebra $mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $Lambda(V,W)$ and $Lambda^infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.
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