Cluster automorphisms and hyperbolic cluster algebras
Cluster automorphisms and hyperbolic cluster algebras
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2012
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通讯作者：
Ibrahim A Saleh
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作者：
Ibrahim A Saleh
设An(S)是域K上的系数自由交换簇代数，簇自同构是Aut.Kk(t1，···，tn)的一个元素，它使得所有簇变量的集合χS保持不变.在第二章中，利用域K(t1，···，tn)的所有种子的集合上的对称群作用的轨道来研究所有这类自同构群。在第三章中，我们建立了一类具有非对易簇结构的新的非对易代数。这种结构与一些双曲代数如Weyl代数、SL2的经典和量子化泛包络代数以及SL(2)的量子坐标代数有着天然的联系。这种簇结构产生了一些组合数据，称为簇串，用来引入Weyl代数的一类表示。同样的数据也引入了不可约和不可分解的表示。第三章的最后一节介绍了一类具有双曲簇结构的范畴。这些范畴的例子是某些代数的表示范畴，例如Weyl代数、李代数SL2的坐标代数和SL(2)的量子坐标代数。由易卜拉欣·A·萨利赫B.A.，开罗大学1995年硕士，开罗大学2002年硕士，堪萨斯州立大学2008年提交的部分满足数学文理学院哲学博士学位要求的论文，堪萨斯州立大学曼哈顿，堪萨斯，2012
Let An(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.KK(t1, · · · , tn) which leaves the set of all cluster variables, χS, invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t1, · · · , tn). In Chapter 3, we set up for a new class of non-commutative algebras that carry a noncommutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl2, and the quantum coordinate algebra of SL(2). CLUSTER AUTOMORPHISMS AND HYPERBOLIC CLUSTER ALGEBRAS by IBRAHIM A SALEH B.A., Cairo University 1995 M.S., Cairo University 2002 M.S., Kansas State University 2008 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2012