**Transformation Groups Seminar**

Speaker: Li Cai (Xi'an Jiaotong-Liverpool University)

"Generators of the commutator of a graph product of connected Hopf algebras over a field"

Time: 10:30 - 11:30

Room: MC 108

In this talk we generalized the result of J. Grabic, T. Panov, S. Theriault and J. Wu on the generators of the homology (endowed with the Pontryagin product) of the loop space of a moment-angle complex associated to a flag complex. Let $A$ be a graph product of connected Hopf algebras $A_1$,â€¦, $A_m$ over a field, and let $Aâ€™â€™$ be the tensor product of $A_1$,â€¦, $A_m$ namely the abelianization of $A$. We consider the bar construction of $BA$ as a chain complex over $k$, and show that it can be reduced to the polyhedral product of $BA_1$, â€¦, $BA_m$, in the category of chain complexes over $k$. Then the generators of the commutator $Aâ€™=\ker(A \to Aâ€™â€™)$ comes from a combinatorial description of the $Tor$ functor on $Aâ€™$, together with the action of $Aâ€™â€™$ on it.