Noncommutative Algebraic Geometry and Noncommutative Invariant Theory

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This research project concerns noncommutative algebra with connections to both invariant theory and algebraic geometry. An algebra is a fundamental object throughout mathematics. One use of algebras is to encode information about geometric structures. Many algebras arising in mathematics and physics are noncommutative, i.e., the product of two elements depends on the order in which elements are multiplied. The PI will investigate noncommutative algebras that are related to noncommutative geometry and noncommutative invariant theory.The research of the principal investigator lies in two subfields of noncommutative algebra entitled Noncommutative Projective Algebraic Geometry (NCPAG) and Noncommutative Invariant Theory (NCIT). The first area was launched in the 1980s to examine algebras, especially the three-dimensional Sklyanin algebras Skly3, whose ring-theoretic behavior could not be determined using purely algebraic techniques. The PI plans to continue using techniques of NCPAG to establish results on Skly3 and on algebras arising in physics and Lie theory, such as the Virasoro algebra and other related algebras. The goal of the second area of research, NCIT, is to extend results in classical invariant theory to a noncommutative setting. To do so, we replace an action of a group on a commutative polynomial ring by an action of a Hopf algebra (or quantum group) on a noncommutative regular algebra that shares homological properties with its commutative counterpart. Although it is difficult for a Hopf algebra to act on an algebra, the PI has works in progress on actions of Hopf algebras on commutative domains, on their quantizations, and on algebras arising from NCPAG. The PI also intends to continue to contribute results on algebras arising from these actions, such as invariant subrings and smash product algebras. This will lead to new examples of algebras whose representation theory merit further investigation.