Hopf Algebra: An Introduction (Chapman & Hall/CRC Pure and by Sorin Dascalescu, Constantin Nastasescu, Serban Raianu

By Sorin Dascalescu, Constantin Nastasescu, Serban Raianu

This research covers comodules, rational modules and bicomodules; cosemisimple, semiperfect and co-Frobenius algebras; bialgebras and Hopf algebras; activities and coactions of Hopf algebras on algebras; finite dimensional Hopf algebras, with the Nicholas-Zoeller and Taft-Wilson theorems and personality conception; and extra.

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Example text

Moreover, if all the spaces M~are finite dimensional, then ~ is an isomorphism. Proof: The assertion follows immediately by induction from asertion iii) of the lemma. | If X, Y are k-vector spaces and v : X -~ Y is a k-linear map, we will denote by v* : Y* --~ X* the map defined by v*(f) fv for any f ~ Y*. Wemade all the necessary preparations for constructing the dual algebra of a coalgebra. Let then (C,A,z) be a coalgebra. 2, and u : k --~ C*,u -- ¢*¢~ where ¢ : k --~ k* is the canonical isomorphism.

Then: i) If S is a subalgebra in C*, then ± i s a coideal in C. ii) If I is a coideal in C, then ± i s a subalgebra inC*. 5. 47 THE FINITE DUAL Proof: i) Let i : S --~ C* be the inclusion, which is a morphismof algebras. Then i ° : C*° -~ S° is a morphism of coalgebras, and hence if ¢c : C -~ C*° is the canonical coalgebra map, we have a morphism of coalgebras i°¢c :C~ S ° . Thus Ker(i°¢c) = S ±. Since the kernel of a coalgebra mapis. a coideal, assertion i) is proved. ii) Let v: : C -~ C/I be the canonical projection, which is a coalgebra map.

7 The algebra C* defined above is called the dual algebra of the coalgebra C. The multiplication of C* is called convolution. Most of the times (if there is no danger of confusion), we will simply write f g instead of f * g for the convolution product of f and g. 4 1). Then the dual algebra is (kS)* = Horn(kS, k) with multiplication defined by (f ¯ g)(s) = :(s)g(s) for f, g E (kS)*, s ~ S. Denoting by Map(S, k) the algebra of functions from S to k, the map 0 : (kS)* -~ Map(S, k) associating to a morphism f ~ (kS)* its restriction to S is an algebra isomorphism.

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