Hopf–Galois Systems and Kashiwara Algebras

Abstract

This article is made up with two parts. In the first part, using a recent result of Schauenburg, one generalizes to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf–Galois objects and Hopf–Galois systems. In this last description, one gives explicitly an inverse for a Hopf–Galois object T together with its generalized antipode. In the second part of the article, one shows that the Kashiwara algebras introduced by Kashiwara in his study of crystal bases form Hopf–Galois systems under the coaction of a quantized enveloping algebra of a Kac–Moody algebra. Their classical limits are examples of Sridharan algebras.

Key Words:

• Hopf–Galois system
• Hopf–Galois extension
• Kashiwara algebra
• Sridharan algebra
• Torsor

Keywords:

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Acknowledgments

I am very grateful to B. Enriquez for his help and support. I am also grateful to J. Bichon, P. Baumann and P. Schauenburg for remarks. I would also like to thank W. Soergel for the hospitality of the Freiburg University.

Notes

^#Communicated by J. Alev.