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COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE
, 2002
"... Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general ab ..."
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Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by
On the cohomology of a smash product of Hopf algebras, preprint
, 2002
"... A five term sequence for the low degree cohomology of a smash product of (cocommutative) Hopf algebras is obtained, generalizing that of Tahara for a semidirect product of groups 0 ..."
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Cited by 2 (1 self)
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A five term sequence for the low degree cohomology of a smash product of (cocommutative) Hopf algebras is obtained, generalizing that of Tahara for a semidirect product of groups 0
On Linear Difference Equations over Rings and Modules
, 2003
"... In this note we develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground ..."
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Cited by 1 (0 self)
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In this note we develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.
Hopf Pairings and Induction Functors over Rings
, 2003
"... The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for ..."
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The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings), satisfying the so called αcondition. For those pairings the induction functor is studied and a nice description of it is obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (noetherian) ground rings.
Hopf Pairings and (Co)induction Functors over Commutative Rings
, 2004
"... (Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine gr ..."
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(Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called αcondition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings.