Integral Bases of Cluster Algebras and Representations of Tame Quivers
نویسندگان
چکیده
In [CK] and [SZ], the authors constructed the bases of cluster algebras of finite types and of type e A1,1, respectively. In this paper, we will deduce Z-bases for cluster algebras of affine types and compare Z-bases with generic variables in [Du2]. Moreover, we give inductive formulas for computing the multiplication between two generalized cluster variables associated to objects in a tube.
منابع مشابه
Tame Quivers and Affine Enveloping Algebras
Let g be an affine Kac-Moody algebra with symmetric Cartan datum, n be the maximal nilpotent subalgebra of g. By the Hall algebra approach, we construct integral bases of the Z-form of the enveloping algebra U(n). In particular, the representation theory of tame quivers is essentially used in this paper.
متن کاملThe Multiplication Theorem and Bases in Finite and Affine Quantum Cluster Algebras
We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in [19]. Moreover some ZP-bases in quantum cluster algebras of finite and affine types are constructed. Under the specialization q and coefficients to 1, these bases are the integral bases of cluster algebra of finite and affine type...
متن کاملOn minimal disjoint degenerations of modules over tame path algebras
For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0 → U → M → V → 0 with indecomposable ends that add up to N . We study these ’building blocs’ of...
متن کاملQuantized Chebyshev polynomials and cluster characters with coefficients
We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type A. ...
متن کاملA Universal Investigation of $n$-representations of $n$-quivers
noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $math...
متن کامل