On Hopf algebras with nonzero integral

Speaker: Juan Cuadra, University of Almeria. 

The Haar measure on a compact group induces a linear functional s on the Hopf algebra of its representative functions. The invariance property of the Haar measure reads as a condition on s that can be expressed in Hopf algebraic terms. Sweedler defined an algebraic notion of integral for Hopf algebras using this condition. Hopf algebras having a nonzero integral are also called co-Frobenius. The quantized coordinate algebra O_q(G) of a simple algebraic group G is one of the most relevant examples of co-Frobenius Hopf algebras. When q is a root of unity, the usual coordinate algebra O(G) is a central Hopf subalgebra of O_q(G) and O_q(G) is finitely generated as a module over O(G). Moreover, O}(G) coincides with the Hopf socle of O_q(G).

Based on this example, Andruskiewitsch and Dascalescu asked (see Co-Frobenius Hopf algebras and the coradical filtration, Math. Z. 243 (2003), 145-154) if any co-Frobenius Hopf algebra is finitely generated as a module over its Hopf socle.

In this talk we will introduce a new family of co-Frobenius Hopf algebras that answers in the negative this question. The results that will be presented are part of a joint work with N. Andruskiewitsch (National University of Cordoba, Argentina) and P. Etingof (Massachusetts Institute of Technology, USA). Arxiv:1206.5934.