# 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional by Boyer Ch. P.

By Boyer Ch. P.

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D From now on we assume that A is finitely generated and projective with finite dual basis {a,, a* | i = 1, • • • , m}. The proof of the next Lemma is straightforward, and therefore left to the reader. 6 Let (A, C, i]j) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. ($)CA. The structure is given by the formulae pr(a*®c] = I/ * ,~~ \ _ p \Q> 09 Cj ^ a* c (1) <8>C( 2 ), (52) / * \ y^ /o * /o\ T^ \d , &iih )C( -\\ 09 a^ Qv C / r j \ . Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved. 44 Bueso et al. 12 The following conditions are equivalent for a k~algebra R. (i) R is filtered by a finite-dimensional filtration with semi-commutative associated graded algebra. (ii) R is filtered with semi-commutative associated graded algebra. -bounded quantum relations for some admissible order X. (iv) R satisfies a set Q of ^ (ii) is obvious. (1). (iii) => (iv). 8 (iv) =>• (i). This gives (46). Conversely, if$ € V{ and e e VFj satisfy (46), then application of (46) to the second and third factors in TOpi ® mli} ® 1> an<^ then EC to the second factor shows that (48) holds for all M e M(t/>)%. Finally note that (48) is equivalent to (2). ) is a Frobenius pair, when is A finitely generated projective as a fc-module. We give a partial answer in the next Proposition. We assume that ip is bijective (cf. [2, Section 6]). In the Doi-Hopf case, this is true if the underlying Hopf algebra H has a twisted antipode.

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