Collections > UNC Chapel Hill Undergraduate Honors Theses Collection > Non-Commutative Quiver Algebras and Their Geometric Realizations
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The main goal of this paper is to study the (commutative) geometric objects whose properties correspond to those of the noncommutative algebras. Specically we are interested in the direct deformations of the usual commutative polynomial ring, and we use the tool of quiver algebras and representations to construct vector bundles Ei on projective spaces Pn whose endormorphism ring End(∑i Ei) corresponds to these deformations of the polynomial ring. These bundles will turn out to be the symmetric powers of T (-1) when the algebra is chosen to be the usual polynomial ring. We also derive some properties and exact sequences of these bundles, and in particular there is a short exact sequence showing that Symk(T (-1)) can be generated by O and O(-1) in the derived category on Pn for any k <=n.