Symmetries and quantum symmetries of Artin-Schelter regular algebras

Classical invariant theory studies the invariants under actions of finite subgroups \$G \subseteq \mathrm{GL}_n(\mathbb{C})\$ on \$\mathbb{C}[x_1, \dots, x_n]\$. In this talk we consider replacing \$\mathbb{C}[x_1, \dots, x_n]\$ by a noncommutative Artin-Schelter regular algebra \$A\$ (a connected graded algebra with certain homological properties of commutative polynomial rings, for example the skew polynomials \$\mathbb{C}_{q_{ij}}[x_1, \dots, x_n]\$, where \$x_i x_j = q_{i,j} x_jx_i\$ for \$0\neq q_{i,j} \in \mathbb{C}\$). We consider both group actions (symmetries) and semisimple Hopf actions (quantum symmetries) on \$A\$ and their invariant subrings \$A^H\$.