Cluster automorphisms and hyperbolic cluster algebras
dc.contributor.author | Saleh, Ibrahim A. | |
dc.date.accessioned | 2012-08-15T18:33:11Z | |
dc.date.available | 2012-08-15T18:33:11Z | |
dc.date.graduationmonth | August | en_US |
dc.date.issued | 2012-08-15 | |
dc.date.published | 2012 | en_US |
dc.description.abstract | Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s 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(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 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 sl[subscript]2, and the quantum coordinate algebra of SL(2). | en_US |
dc.description.advisor | Zongzhu Lin | en_US |
dc.description.degree | Doctor of Philosophy | en_US |
dc.description.department | Department of Mathematics | en_US |
dc.description.level | Doctoral | en_US |
dc.identifier.uri | http://hdl.handle.net/2097/14195 | |
dc.language.iso | en_US | en_US |
dc.publisher | Kansas State University | en |
dc.subject | Cluster Algebras | en_US |
dc.subject | Representation theory | en_US |
dc.subject | Hyperbolic algebras | en_US |
dc.subject.umi | Mathematics (0405) | en_US |
dc.title | Cluster automorphisms and hyperbolic cluster algebras | en_US |
dc.type | Dissertation | en_US |