Hopf algebras: an introduction

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All right, what about the book under review? My point is made: these are pretty austere and imposing beasts, and there is no fast-and-easy way to get at them. Here Hopf algebras play a dual role: first the left invariant differential operators on an algebraic group form a cocommutative Hopf algebra, which coincides with the enveloping algebra of the Lie algebra in characteristic 0, but not in characteristic p. Second: the regular functions on an affine algebraic group, under ordinary multiplication, form a commutative Hopf algebra.

So there is an awful lot going on! With this as the context for Hopf algebras, what about the book under review? This said, a caveat is in order. With the aforementioned definition occurring on p. Beyond this, the reader is certainly well-advised to hit the exercises in the book with zeal and commitment, even if he has a decent background in commutative algebra, algebraic geometry and category theory. In any case, once Hopf algebras have been properly introduced and the whole affair gets off the ground, the focus falls on Hopf orders cf.


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The Spectrum of a Ring. The Zariski Topology on the Spectrum. Representable Group Functors. Hopf Algebras. Larson Orders. Knots and Primes. Masanori Morishita.


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    HOPF Algebras: An Introduction

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