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A Primer of Hopf Algebras

In: Frontiers in Number Theory, Physics, and Geometry II

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  • Pierre Cartier

    (Institut Mathématique de Jussieu/CNRS)

Abstract

In this paper, we review a number of basic results about so-called Hopf algebras. We begin by giving a historical account of the results obtained in the 1930's and 1940's about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with Milnor-Moore's theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions and the corresponding multiple zeta values.

Suggested Citation

  • Pierre Cartier, 2007. "A Primer of Hopf Algebras," Springer Books, in: Pierre Cartier & Pierre Moussa & Bernard Julia & Pierre Vanhove (ed.), Frontiers in Number Theory, Physics, and Geometry II, pages 537-615, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-30308-4_12
    DOI: 10.1007/978-3-540-30308-4_12
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