Download e-book for kindle: Algebras, Rings and Modules: Lie Algebras and Hopf Algebras by Michiel Hazewinkel

Algebraic Geometry

By Michiel Hazewinkel

The most aim of this ebook is to offer an creation to and functions of the idea of Hopf algebras. The authors additionally speak about a few vital facets of the idea of Lie algebras. the 1st bankruptcy might be considered as a primer on Lie algebras, with the most aim to give 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 particularly to offer enough history on those subject matters to be used mostly a part of the e-book. Chapters 4-7 are dedicated to 4 of the main appealing Hopf algebras presently identified: the Hopf algebra of symmetric capabilities, the Hopf algebra of representations of the symmetric teams (although those are isomorphic, they're very diversified 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 diversifications. The final bankruptcy is a survey of functions of Hopf algebras in lots of diverse components of arithmetic and physics. particular positive factors of the publication comprise a brand new option 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 points of the entire Hopf algebras pointed out, emphasis at the combinatorial facets of Hopf algebras, and a survey of functions already pointed out. The e-book additionally comprises an in depth (more than seven hundred entries) bibliography.

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3. Give an example of a surjective local homeomorphism that is not a covering. 4. * Let f : X −→ Y be a local homeomorphism, where X is compact. ) covering with finite fibers. Let X be a G-space, where G is a (discrete) group. For a subgroup H of G, define X H = {x|hx = x for all h ∈ H} ⊂ X; X H is the H-fixed point subspace of X. Topologize the set of functions G/H −→ X as the product of copies of X indexed on the elements of G/H, and give the set of G-maps G/H −→ X the subspace topology. 5. Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H .

We generally write B ∪g X for the pushout of a given cofibration i : A −→ X and a map g : A −→ B. 43 44 COFIBRATIONS Lemma. If i : A −→ X is a cofibration and g : A −→ B is any map, then the induced map B −→ B ∪g X is a cofibration. Proof. Notice that (B ∪g X)×I ∼ = (B ×I)∪g×id (X ×I) and consider a typical test diagram for the HEP. The proof is a formal chase of the following diagram: A qq qq g qq qq qq 5 i0 G A×I o o o g×idoo ooo o o ow G B×I r r h rr rrr r r rx r pushout pushout i×id i Yw Yy1 eu u w u ˜ f ww h u 1 w uu ww  1  ww † † G (B ∪g X) × I B ∪g X † † gxx wY † † ww † † xxxxx w † † ww ¯ † †xxxx  h  www † G X × I.

However, our interest here is in discrete groups G, for which the continuity condition just means that action by each element of G is a homeomorphism. The functoriality on O(G) of our construction of general covers will be immediate from the following observation. Lemma. Let X be a G-space. Then passage to orbit spaces defines a functor X/(−) : O(G) −→ U . Proof. The functor sends G/H to X/H and sends a map α : G/H −→ G/K to the map X/H −→ X/K that sends the coset Hx to the coset Kγ −1 x, where α is given by the subconjugacy relation γ −1 Hγ ⊂ K.

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