Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

Thomas Timmermann, and Alfons Van Daele. "Multiplier Hopf algebroids arising from weak multiplier Hopf algebras." Banach Center Publications 106.1 (2015): 73-110. <http://eudml.org/doc/282187>.

@article{ThomasTimmermann2015,
abstract = { It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra. },
author = {Thomas Timmermann, Alfons Van Daele},
journal = {Banach Center Publications},
keywords = {bialgebroids; multiplier Hopf algebroids; weak Hopf algebras; weak multiplier Hopf algebras; quantum groupoids},
language = {eng},
number = {1},
pages = {73-110},
title = {Multiplier Hopf algebroids arising from weak multiplier Hopf algebras},
url = {http://eudml.org/doc/282187},
volume = {106},
year = {2015},
}

TY - JOUR
AU - Thomas Timmermann
AU - Alfons Van Daele
TI - Multiplier Hopf algebroids arising from weak multiplier Hopf algebras
JO - Banach Center Publications
PY - 2015
VL - 106
IS - 1
SP - 73
EP - 110
AB - It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.
LA - eng
KW - bialgebroids; multiplier Hopf algebroids; weak Hopf algebras; weak multiplier Hopf algebras; quantum groupoids
UR - http://eudml.org/doc/282187
ER -