Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Author "Berenstein, Arkady"

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    Factorizable Module Algebras, Canonical Bases, and Clusters
    (University of Oregon, 2018-09-06) Schmidt, Karl; Berenstein, Arkady
    The present dissertation consists of four interconnected projects. In the first, we introduce and study what we call factorizable module algebras. These are $U_q(\mathfrak{g})$-module algebras $A$ which factor, potentially after localization, as the tensor product of the subalgebra $A^+$ of highest weight vectors of $A$ and a copy of the quantum coordinate algebra $\mathcal{A}_q[U]$, where $U$ is a maximal unipotent subgroup of $G$, a semisimple Lie group whose Lie algebra is $\mathfrak{g}$. The class of factorizable module algebras is surprisingly rich, in particular including the quantum coordinate algebras $\mathcal{A}_q[Mat_{m,n}]$, $\mathcal{A}_q[G]$ and $\mathcal{A}_q[G/U]$. It is closed under the braided tensor product and, moreover, the subalgebra $A^+$ of each such $A$ is naturally a module algebra over the quantization of $\mathfrak{g}^*$, the Lie algebra of the Poisson dual group $G^*$. The aforementioned examples of factorizable module algebras all possess dual canonical bases which behave nicely with respect to factorization $A=A^+\otimes \mathcal{A}_q[U]$. We expect the same is true for many other members of this class, including braided tensor products of such. To facilitate such a construction in tensor products, we propose an axiomatic framework of based modules which, in particular, vastly generalizes Lusztig's notion of based modules. We argue that all of the aforementioned $U_q(\mathfrak{g})$-module algebras (and many others) with their dual canonical bases are included, along with their tensor products. One of the central objects of study emerging from our generalization of Lusztig's based modules is a new (very canonical) basis $\mathcal{B}^{\diamond n}$ in the $n$-th braided tensor power $\mathcal{A}_q[G/U]$. We argue (yet conjecturally) that $\mathcal{A}_q[G/U]^{\underline{\otimes}n}$ has a quantum cluster structure and conjecture that the expected cluster structure structure on $\mathcal{A}_q[G/U]^{\underline{\otimes}n}$ is completely controlled by the real elements of our canonical basis $\mathcal{B}^{\diamond n}$. Finally, in order to partially explain the monoidal structures appearing above, we provide an axiomatic framework to construct examples of bialgebroids of Sweedler type. In particular, we describe a bialgebroid structure on $\mathfrak{u}_q(\mathfrak{g})\rtimes\mathbb{Q} C_2$, where $\mathfrak{u}_q(\mathfrak{g})$ is the small quantum group and $C_2$ is the cyclic group of order two. This dissertation contains previously published co-authored material.
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    Quantum Cluster Characters
    (University of Oregon, 2012) Rupel, Dylan; Rupel, Dylan; Berenstein, Arkady
    We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations are related to re ection functors and internal mutations are related to tilting theory. Our main result will show the quantum cluster character gives a cluster monomial in this quantum cluster algebra whenever the representation is rigid, moreover we will see that each non-initial cluster variable can be obtained in this way from the quantum cluster character.
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    Semisimplicity of Certain Representation Categories
    (University of Oregon, 2013-10-03) Foster, John; Berenstein, Arkady
    We exhibit a correspondence between subcategories of modules over an algebra and sub-bimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence of a Peter-Weyl decomposition of the corresponding sub-bimodule. Finally, we use this technique to establish the semisimplicity of certain finite-dimensional representations of the quantum double $D(U_q(sl_2))$ for generic $q$.