General definition in a coring
Given an -coring (a comonoid in the category of --bimodules for a -algebra ) (example: any -algebra) a semi-grouplike element in is any such that
A grouplike (or group-like) element is a semi-grouplike one such that .
Proposition. An -coring has a grouplike element iff is a right or left -comodule.
Proof. Given a grouplike element , one defines a right coaction by the formula
it is clear that this is a map of -bimodules. Now , while hence the coassociativity and similarly for the counit.
Conversely, let be a right -comodule. Then one checks that is a grouplike.
For the left comodules the story is similar, e.g. .
Special case: grouplike elements in coalgebras
Every coalgebra is special case of a coring.
The grouplike elements in a -Hopf algebra form a group. (Can this fact be categorified ??)
Relation to differential graded algebras
For corings with a (sometimes semi-)grouplike element one can define many useful notions which do not exist for general corings.
For example, given a semi-grouplike element , the tensor algebra of the coring , where ( times) over can be equipped with a differential of degree in a canonical way making it a differential graded algebra:
in degree , one defines
and in higher degree
In fact, by a result in
- A. V. Roiter, Matrix problems and representations of BOCS’s; in Lec. Notes. Math. 831, 288–324 (1980)
semi-free differential graded algebras are in bijective correspondence with corings with a group-like element. Moreover flat connections for a semi-free dga are in - correspondence with the comodules over the corresponding coring with a group-like element.
A special case of this construction is when and is the Sweedler coring for a -algebra extension . The dga obtained is the classical Amitsur complex for that extension; for this reason the complex above for any coring and semi-grouplike is sometimes said to be an Amitsur complex.
T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
C. Menini, D. Ştefan, Descent theory and Amitsur cohomology of triples, J. Algebra 266 (2003), no. 1, 261–304.
T. Brzeziński, Flat connections and comodules, math.QA/0608170
T. Brzeziński, Galois structures, Warszawa 2007/8 course, part III, pdf, ps