Cluster automorphisms and hyperbolic cluster algebras

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Show simple item record Saleh, Ibrahim A. 2012-08-15T18:33:11Z 2012-08-15T18:33:11Z 2012-08-15
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.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.title Cluster automorphisms and hyperbolic cluster algebras en_US
dc.type Dissertation en_US Doctor of Philosophy en_US
dc.description.level Doctoral en_US
dc.description.department Department of Mathematics en_US
dc.description.advisor Zongzhu Lin en_US
dc.subject.umi Mathematics (0405) en_US 2012 en_US August en_US

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