By E. R Kolchin
ISBN-10: 0124176402
ISBN-13: 9780124176409
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29 below). The same argument provides a new proof for the Fundamental Theorem of Coalgebras (Theorem 1. 3. 7). 17 If C is a coalgebra, show that C is cocommutative if and only if C* is commutative. 18 Let A be an algebra. Then the map iA : °*, A --~ A defined by iA(a)(a*) = a*(a) for any a ~ A,a* °, is a morphism of algebras. Proof: Wehave first thatiA(1)(a*) = a*(1) eAo(a*), soiA(1) = edO, which is the identity of A°*. Then for any a,b ~ A,a* ~ A° we have 44 CHAPTER1. ALGEBRASAND COALGEBRAS iA(ab)(a*) = a*(ab), and (iA(a)iA(b))(a*) = EiA(a)(a;)iA(b)(b;) P = E a~(a)b;(b) P --- (¢A(a*))(a®b) = M*(a*)(a®b) -- (a*M)(a®b) = a*(ab) where we have denoted iA(ab) = iA(a) .
Thus n >_ 2. Then all a~ are non-zero (otherwise we would have such a linear combination for smaller n). , g,~ are linearly independent (otherwise again we would obtain one of them as a linear combination of less then n grouplike elements), it follows that for i ~ j we have aiaj = 0, a contradiction. If A is a finite dimensional algebra, then the grouplike elements of the dual coalgebra have a special meaning. 15 Let A be a finite dimensional algebra and A* the dual coalgebra of A. Then G(A*) = Alg(A, k), the algebra maps from A to Proof: Let f ~ A*.
Then (E~(a~)b~)(a)=~a~(1)b~(a)=a*(1. i i for any a E A, and therefore ~-:~i e(ai)* b* and the proof is complete. = a*. a morphism of algebras. Then f*(B °) C_ A° and the induced map f° : B° ~ A° is a morphism of coalgebras. 2 that f*(B °) C_ A°. In order to show that f° is a morphism of coalgebras, we have to prove that the following two diagrams are commutative. BO B° °® B fO fO fo®fo . = b*(1) °. for any b* ~ B As for the first diagram, let b* ~ B°, Aso(b*) = ~-~i b~ ®c~’. Since ¢ : A°® A° --~ (A ® A)* is injective, in order to show that (f° ® f°)ABO= /kAof° it is enough to show that ¢(fo® f°)ABO ¢AAof°.
Differential algebraic groups by E. R Kolchin
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