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On the Algebra of Operations for Hopf Cohomology
Author(s) -
Singer William M.
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004297
Subject(s) - mathematics , hopf algebra , steenrod algebra , algebra over a field , coproduct , quasitriangular hopf algebra , bialgebra , cohomology , pure mathematics , mathematics subject classification , multiplication (music) , representation theory of hopf algebras , subject (documents) , filtered algebra , combinatorics , division algebra , computer science , library science
In his thesis ( Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations Sq k on the cohomology ring of any cocommutative Hopf algebra over Z /2. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that Sq 0 is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra B that is generated by elements Sq k and subject to Adem relations; it shows that the Cartan formula gives a well‐defined coproduct on B . Also, it is shown that B with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper. 2000 Mathematics Subject Classification 55S10 (primary), 55T15 (secondary).