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Алгебра и анализ Том 7 (1995), вып. 1
© 1995 г.
EXTENSIONS OF HOPF ALGEBRAS
N. Andruskiewitsch, J. Devoto Abstract. We investigate the notion of exact sequences of Hopf algebras. To two Hopf algebras A and B, and a data consisting of an action of В on A, a cocycle, a coaction of A on B, and a co-cocycle we associate a short exact sequence of Hopf algebras 0 —• A —• С -* В —• 0. We define cleft short exact sequences of Hopf algebras and prove that their isomorphism classes are in a bijective correspondence with the quotient set of datas as above such that the cocycle and the co-cocycle are invertible, modulo a natural action of a subgroup of Reg(B, A).
§0. Introduction
The paper deals with extensions of Hopf algebras. By definition [Dl], the category of quantum groups is the dual category to the category of Hopf algebras with bijective antipode. For this reason we can work mainly in this second category and all the results will translate to that of quantum groups in the obvious way. (Some of the results below are true with a weaker hypothesis about the antipode). It helps our intuition, however, to keep in mind that a Hopf algebra is the "algebra of functions on a quantum group". Let us fix, for simplicity, a commutative field к and let us briefly say "Hopf algebra" for a Hopf algebra over k. We shall use the following notation: m, A (or 5), e, S mean respectively the multiplication, comultiplication, counit, antipode of a Hopf algebra (or an algebra or a coalgebra), specified with a subscript if necessary. The opposite (cp)multiplication is betokened by a superscript "op". We shall also use the following convention: if с is an element of a tensor product A B, then we write с = с,- ® с*, omitting the summation symbol. An exception is the case с = А(х), where we use Sweedler's "sigma" notation but dropping again the summatory. The usual transposition NN' —> N'giN is denoted by T.lig: N®N -» N®N is a morphism offc-modulesthen gij. ft®m _^ jy®m j , ^ m e u s u a j meaning, for example, gi,i+1 — idNi_l ®g
