By Michiel Hazewinkel

The most objective of this booklet is to give an advent to and purposes of the speculation of Hopf algebras. The authors additionally speak about a few very important points of the speculation of Lie algebras. the 1st bankruptcy could be considered as a primer on Lie algebras, with the most aim to provide an explanation for and turn out the Gabriel-Bernstein-Gelfand-Ponomarev theorem at the correspondence among the representations of Lie algebras and quivers; this fabric has no longer formerly seemed in publication shape. the subsequent chapters also are ''primers'' on coalgebras and Hopf algebras, respectively; they target in particular to provide enough historical past on those subject matters to be used normally a part of the booklet. Chapters 4-7 are dedicated to 4 of the main appealing Hopf algebras at the moment recognized: the Hopf algebra of symmetric services, the Hopf algebra of representations of the symmetric teams (although those are isomorphic, they're very varied within the facets they create to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric features (these are twin and either generalize the former two), and the Hopf algebra of diversifications. The final bankruptcy is a survey of purposes of Hopf algebras in lots of different components of arithmetic and physics. certain gains of the e-book comprise a brand new strategy to introduce Hopf algebras and coalgebras, an intensive dialogue of the numerous common homes of the functor of the Witt vectors, an intensive dialogue of duality elements of all of the Hopf algebras pointed out, emphasis at the combinatorial features of Hopf algebras, and a survey of functions already pointed out. The publication additionally comprises an intensive (more than seven hundred entries) bibliography

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**Algebras, rings, and modules : Lie algebras and Hopf algebras**

The most aim of this ebook is to provide an advent to and purposes of the idea of Hopf algebras. The authors additionally speak about a few very important elements of the speculation of Lie algebras. the 1st bankruptcy could be seen as a primer on Lie algebras, with the most target to provide an explanation for and end up the Gabriel-Bernstein-Gelfand-Ponomarev theorem at the correspondence among the representations of Lie algebras and quivers; this fabric has no longer formerly seemed in ebook shape.

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**Additional resources for Algebras, rings, and modules : Lie algebras and Hopf algebras**

**Example text**

2) A be a discrete subring of a local field K with Let F C K. We assume that (1) A K/A is a compact abelian group. = :It C F = Q C K = IR. C F = lFq(t) = lFq(t- ) C K = IFq«t- (4) A = IFq [C - curve and 00] C F Foo = IFq (C - (0) = IFq (C) C K is the completion of l = Foo F at 00 FC K =C • ». where C is an affine • Of course (3) is a special case of (4),and (4) is the case of interest in this part. given by a norm. exists a neighborhood then Nt of A subgroup 0 in V over H in V with K, the topology is well V is discrete provided there N' n H = O.

Ker(u)(k) u =0 ~a. vu always exists for that u: w(ker(u)(k» = ~ --+-

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