nLab grouplike element

Definition

General definition in a coring

Given an $A$-coring $(C,\Delta,\epsilon)$ (a comonoid in the category of $A$-$A$-bimodules for a $k$-algebra $A$) (example: any $k$-algebra) a semi-grouplike element in $A$ is any $g\in C$ such that

$$\Delta(g) = g\otimes g \,.$$

A grouplike (or group-like) element is a semi-grouplike one such that $\epsilon(g) = 1$.

Proposition. An $A$-coring $(C,\Delta,\epsilon)$ has a grouplike element iff $A$ is a right or left $C$-comodule.

Proof. Given a grouplike element $g\in C$, one defines a right coaction $\rho = \rho_g : A \to A\otimes_A C\cong C$ by the formula

$$\rho(a) = 1_A \otimes_A ga = ga$$

it is clear that this is a map of $A$-bimodules. Now $(\rho\otimes id_C)(1_A\otimes_A ga) = g\otimes_A 1_A\otimes ga = g\otimes ga$, while $(id\otimes \Delta_C)(1_A\otimes ga) = 1_A \otimes_A g\otimes ga = g\otimes ga$ hence the coassociativity and similarly for the counit.

Conversely, let $(A,\rho)$ be a right $C$-comodule. Then one checks that $\rho(1_A)\in A\otimes_A C \cong C$ is a grouplike.

For the left comodules the story is similar, e.g. $\rho(a) = ag$.

Note that coring homomorphisms send grouplike elements to grouplike elements. This can be assembled into a functor, and moreover, this functor is representable.

Theorem

The functor $G : A\text{-}\mathsf{CoRing} \to \mathsf{Set}$ is representable: $G \cong \mathrm{Hom}_{A\text{-}\mathsf{CoRing}}(A,-)$.

Proof.

We can assign each $g \in GC$ the map $\rho_g$ by $1_A \mapsto g$. It is easily checked that this is indeed a map of coalgebras, and the inverse is to take the image of $1_A$. The assignments are immediately checked to be natural and to be inverses of each other.

Special case: grouplike elements in coalgebras

Every coalgebra is special case of a coring.

The grouplike elements in a $k$-Hopf algebra form a group. Indeed, a Hopf algebra $H$ is just a group over the category of coalgebras over $k$ (the monoidal structure $\otimes$ is even the categorical product), so the set of grouplike elements $\mathrm{Hom}_{k\text{-}\mathsf{CoAlg}}(k,H)$ is canonically a group.

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 $g$, the tensor algebra $\Omega C = \oplus_i \Omega^i C$ of the coring $C$, where $\Omega^i C = C\otimes_A \ldots \otimes_A C$ ($i$ times) over $A$ can be equipped with a differential $d$ of degree $+1$ in a canonical way making it a differential graded algebra:

in degree $0$, one defines

$$d a = g a - a g$$

and in higher degree

$$d(c_1\otimes\ldots\otimes c_n) = g\otimes c_1\otimes\ldots\otimes c_n + (-1)^{n+1}c_1\otimes\ldots\otimes c_n\otimes g + \sum_{i=1}^n c_1\otimes\ldots\otimes c_{i-1}\otimes \Delta(c_i)\otimes c_{i+1}\otimes\ldots \otimes c_n\,.$$

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 $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.

A special case of this construction is when $g = 1\otimes_R 1$ and $C$ is the Sweedler coring for a $k$-algebra extension $R\to S$. The dga obtained is the classical Amitsur complex $\Omega(S/R)$ for that extension; for this reason the complex $\Omega C = \Omega(C,g)$ above for any coring $C$ and semi-grouplike $g$ 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