Noncommutative Symmetric Groups in Noncommutative Projective Algebraic Geometry

In group theory, the symmetric group of a geometric object is the group of all transformations under whose action the object is invariant. It is a principle that the very nature of the symmetric group decodes the fundamental geometry of the space, such as in theoretic physics, the Lorentz transformations reveal Einstein’s famous special theory of relativity in Minkowski spacetime. In this talk, the speaker will extend the principle to noncommutative projective algebraic geometry, where the approach is to bring one of Manin’s universal quantum groups as the noncommutative analogy of the symmetric group of a noncommutative project space defined by an Artin-Schelter (AS) regular algebra. These quantum groups have deep relations with noncommutative invariant theory and quantum symmetry. Many basic concepts and questions will be discussed during the talk together with a recent progress in the case of AS-regular algebras of dimension two. This is a joint work with Chelsea Walton.