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Reflection Hopf Algebras

Ellen Kirkman, Wake Forest University
Tuesday, February 4, 2020 - 2:30pm to 3:30pm
PDL C-38

Let \$\mathbb{k}\$ be an algebraically closed field of characteristic zero. When \$H\$ is a semisimple Hopf algebra that acts inner-faithfully and homogenously on an Artin-Schelter algebra \$A\$, so that the subalgebra of invariants \$A^H\$ is also Artin-Schelter regular, we call \$H\$ a reflection Hopf algebra for \$A\$; when \$H=\mathbb{k}[G]\$ and \$A = \mathbb{k}[x_1, \dots ,x_n]\$ then \$H\$ is a reflection Hopf algebra for \$A\$ if and only if \$G\$ is a reflection group.  We provide examples of reflection groups and reflection Hopf algebras for noncommutative Artin-Schelter algebras.  We show that in this noncommutative context there exist notions of the Jacobian, reflection arrangement, and discriminant that extend the definitions used for reflection groups actions on polynomial algebras.

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