Cluster automorphisms and hyperbolic cluster algebras
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 var...
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ndltd-KSU-oai-krex.k-state.edu-2097-141952017-03-03T15:44:52Z Cluster automorphisms and hyperbolic cluster algebras Saleh, Ibrahim A. Cluster Algebras Representation theory Hyperbolic algebras Mathematics (0405) 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). 2012-08-15T18:33:11Z 2012-08-15T18:33:11Z 2012-08-15 2012 August Dissertation http://hdl.handle.net/2097/14195 en_US Kansas State University |
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Cluster Algebras Representation theory Hyperbolic algebras Mathematics (0405) |
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Cluster Algebras Representation theory Hyperbolic algebras Mathematics (0405) Saleh, Ibrahim A. Cluster automorphisms and hyperbolic cluster algebras |
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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). |
author |
Saleh, Ibrahim A. |
author_facet |
Saleh, Ibrahim A. |
author_sort |
Saleh, Ibrahim A. |
title |
Cluster automorphisms and hyperbolic cluster algebras |
title_short |
Cluster automorphisms and hyperbolic cluster algebras |
title_full |
Cluster automorphisms and hyperbolic cluster algebras |
title_fullStr |
Cluster automorphisms and hyperbolic cluster algebras |
title_full_unstemmed |
Cluster automorphisms and hyperbolic cluster algebras |
title_sort |
cluster automorphisms and hyperbolic cluster algebras |
publisher |
Kansas State University |
publishDate |
2012 |
url |
http://hdl.handle.net/2097/14195 |
work_keys_str_mv |
AT salehibrahima clusterautomorphismsandhyperbolicclusteralgebras |
_version_ |
1718418420663320576 |