Read e-book online Algebras, Rings and Modules: Lie Algebras and Hopf Algebras PDF
By Michiel Hazewinkel
The most target of this ebook is to give an advent to and functions of the idea of Hopf algebras. The authors additionally speak about a few vital points of the idea of Lie algebras. the 1st bankruptcy should 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 now not formerly seemed in publication shape. the following chapters also are "primers" on coalgebras and Hopf algebras, respectively; they target particularly to provide adequate heritage on those themes to be used more often than not a part of the publication. Chapters 4-7 are dedicated to 4 of the main attractive Hopf algebras at present identified: the Hopf algebra of symmetric capabilities, the Hopf algebra of representations of the symmetric teams (although those are isomorphic, they're very varied within the points they create to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric services (these are twin and either generalize the former two), and the Hopf algebra of variations. The final bankruptcy is a survey of purposes of Hopf algebras in lots of assorted elements of arithmetic and physics. specified positive aspects of the booklet comprise a brand new option to introduce Hopf algebras and coalgebras, an in depth dialogue of the various common houses 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 points of Hopf algebras, and a survey of functions already pointed out. The ebook additionally includes an intensive (more than seven hundred entries) bibliography.
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Additional info for Algebras, Rings and Modules: Lie Algebras and Hopf Algebras
An−1 ) with 0 ≤ a ≤ q − 1 for 0 ≤ ≤ n − 1, and n we want to choose an with 0 ≤ an ≤ q − 1 such that =0 a is divisible by n−1 (q − 1). If =0 a is divisible by (q − 1), then we may take an = 0 or an = q − 1; n−1 if a is not divisible by (q − 1), then we have precisely one choice for an . =0 Therefore αn = 2αn−1 + (q n − αn−1 ) = q n + αn−1 . Since α0 = 2, we conclude that αn = (1 + q + . . + q n ) + 1. 3. SUPERSINGULAR CALABI-YAU HYPERSURFACES 47 Therefore dimFq K•F (PnFq ) ≤ |Pn (Fq )| = dimFq K•F (Pn (Fq )).
By applying the theorem for Xm , we get dim(X) (−1)i trace(F m |H i (X, OX )). 2 that we may identify X(Fq ) with the closed points x ∈ X with k(x) = Fq . In what follows we will often make this identification without any further comment. A stronger congruence formula was proved be Deligne [Del2] and Katz [Katz]. In fact, we will also prove a strengthening of the above statement, but in a different direction. The first extension is to sheaves with a Frobenius action. A coherent F -module on X is a coherent sheaf M on X, together with a Frobenius action on M, that is, a morphism of sheaves of OX -modules FM : M → F∗ (M).
20. As a last example, consider the case when X = Spec k, where ´ k is a field. Note that in this case an object in Et(X) is just a disjoint union of finitely many Spec Ki , where the Ki are finite, separable extensions of k. It is clear that every ´etale sheaf F over X is determined by its values MK := F(Spec K), for K/k as above. Furthermore, G(K/k) has an induced action on MK , and for every Galois extension L/K of finite, separable extensions of k, we have a functorial isomorphism MK (ML )G(L/K) .
Algebras, Rings and Modules: Lie Algebras and Hopf Algebras by Michiel Hazewinkel