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Algebraic Topology-Rational Homotopy by Felix Y. (ed.) PDF

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By Felix Y. (ed.)

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All Rights Reserved. 18 Brzeziriski et al. and it follows that

)), as required. D Suppose that C is finitely generated and projective as a /c-module. 7, it follows that F is separable if and only if there exists a map (f> € V^ = Horn ^ A(A <8> C, C* A) such that (/>(! C( 2 ))(c(i)) = e(c)l, for all c e C. In the Doi-Hopf case, this implies the Maschke Theorem in [12]. Now we apply the same procedure to determine when (F, G) is a Frobenius pair. 9 Consider an entwining structure (A, C, ijj), and assume that C is finitely generated projective as a k-module.

4 D CONSEQUENCES AND EXAMPLES Everyone working in the field of non-commutative algebras would greatly appreciate some way to get properties of a given algebra just checking some properties on the defining relations. This is the case for the algebras satisfying a set of quantum relations. You need just to check if the relations are bounded for some admissible order on N™ and the concluding theorems in Section 3, in conjunction with the well developed theory of filtered algebras, will take care of everything.

28 Brzeziriski et al. defined by Let W3 = j(Wi) C B ® B ® A. Take e = bl <8> 62 ® a2 & B ® B ® A (summation implicitely understood). Then e e W$ if and only if (4) holds, for all s = 6#1 and s = l#a with 6 e B and a e A, if and only if bb1 ®b2 ®a2 1 2 (6 )R ® (6 2 ) r (8) aRra = b1 626B ® a^, 1 2 2 = b ® 6 a a, (59) (60) for all a € yl, fe € S. This implies isomorphisms W7 ^ M/i ^ VK3 = {e = b1 62 ® a2 e B ® B & A | (59) and (60) hold}. 2 takes the following form. 1 Let (B, A, R) be a factorization structure over a commutative ring k.

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