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Register a new userA $mathbb{Z}$-basis for the cluster algebra associated to an affine quiver
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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.
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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.
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.
In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra $mathfrak{D}$ and the ideal generated by the squares of elements (further denoted by $I$) is a right $mathfrak{D}$-module. Using description cite{Cas} of representations of algebra $mathfrak{D}$ in $mathfrak{sl}(3,{mathbb{C}})$ and $mathfrak{sp}(4,{mathbb{F}})$ where ${mathbb{F}}={mathbb{R}}$ or ${mathbb{C}}$ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra $mathfrak{D}.$ Classification of Leibniz algebras with corresponding Lie algebra $mathfrak{D}$ and with the ideal $I$ as a Fock right $mathfrak{D}$-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces $BL^2$ and $ZL^2$ are constructed, as well.
In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $mathfrak{e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by $I$) as a right $mathfrak{e}(2)$-module is associated to representations of $mathfrak{e}(2)$ in $mathfrak{sl}_2({mathbb{C}})oplus mathfrak{sl}_2({mathbb{C}}), mathfrak{sl}_3({mathbb{C}})$ and $mathfrak{sp}_4(mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${mathfrak{e(n)}}$ as its liezation $I$ being an $(n+1)$-dimensional right ${mathfrak{e(n)}}$-module defined by transformations of matrix realization of $mathfrak{e(n)}.$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $mathfrak{D}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra $mathfrak{D}_k$ and with the ideal $I$ considered as a Fock $mathfrak{D}_k$-module.
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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