Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / 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).
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/14195 |
| Date | January 1900 |
| Creators | Saleh, Ibrahim A. |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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