Functional Analysis, Linear Structures and Applications Seminar  RSS

19/05/2017, 15:00 — 16:00 — Room 6.2.38, Faculty of Sciences of the Universidade de Lisboa
, Universidade do Porto

Semisimple Hopf actions and factorization through group actions

Let $H$ be a Hopf algebra over a field $F$ acting on an algebra $A$. Let $I \subseteq \operatorname{Ann}_H(A)$ be a Hopf ideal of $H$, then one says that the action of $H$ on $A$ \textit{factors through} the quotient Hopf algebra $H/I$. If there exists $I \subseteq \operatorname{Ann}_H(A)$ such that $H/I \cong F[G]$, for some group $G$, we say that the action of $H$ on $A$ factors through a group action. In 2014, Etingof and Walton have shown that any semisimple Hopf action on a commutative domain factors through a group action [EtingofWalton]. Also in 2014, using their previous result, Cuadra, Etingof and Walton showed that any action of a semisimple Hopf algebra $H$ on the $n$th Weyl algebra $A=A_n(F)$, with $\operatorname{char}(F) = 0$, factors through a group action [CuadraEtingofWalton].

In this talk we will briefly present a generalization of Cuadra, Etingof and Walton's result. Namely, that any action of a semisimple Hopf algebra $H$ on an iterated Ore extension of derivation type in characteristic zero factors through a group action [LompPansera]. We also present a work in progress on semisimple Hopf algebra actions on the quantum polynomial algebras which do not factor through a group actions.

This talk is all based on my upcoming Ph.D. Thesis under the supervision of Christian Lomp.

References

[EtingofWalton] Etingof, P., Walton, C. (2014). Semisimple Hopf actions on commutative domains. Adv. Math. 251: 47–61. doi:10.1016/j.aim.2013.10.008.

[CuadraEtingofWalton] Cuadra, J., Etingof, P., Walton, C. (2015). Semisimple Hopf actions on Weyl algebras Adv. Math. 282:47–55. doi:10.1016/j.aim.2015.05.014.

[LompPansera] Lomp, C., Pansera, D. A note on a paper by Cuadra, Etingof and Walton. Communications in Algebra 1532-4125. doi:10.1080/00927872.2016.1236933

Current organizers: Helena Mascarenhas, Ângela Mestre.

CEAFEL FCT