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grouplike element

Definition

General definition in a coring

Given an $A$-coring $(C,\Delta ,ϵ)$ (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

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

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

Proposition. An $A$-coring $(C,\Delta ,ϵ)$ 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

$$\rho (a)=1_A{\otimes }_A\mathrm{ga}=\mathrm{ga}$$\rho(a) = 1_A \otimes_A ga = ga

it is clear that this is a map of $A$-bimodules. Now $(\rho \otimes {\mathrm{id}}_C)(1_A{\otimes }_A\mathrm{ga})=g{\otimes }_A{1}_A\otimes \mathrm{ga}=g\otimes \mathrm{ga}$, while $(\mathrm{id}\otimes {\Delta }_C)(1_A\otimes \mathrm{ga})=1_A{\otimes }_{A}g\otimes \mathrm{ga}=g\otimes \mathrm{ga}$ 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. $\rho (a)=\mathrm{ag}$.

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. (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 $g$, the tensor algebra $\Omega C={\oplus }_i{\Omega }^{i}C$ of the coring $C$, where ${\Omega }^{i}C=C{\otimes }_A\dots {\otimes }_{A}C$ ($i$ 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

$$da=ga-ag$$d a = g a - a g

and in higher degree

$$d(c_1\otimes \dots \otimes c_n)=g\otimes c_1\otimes \dots \otimes c_n+(-1{)}^{n+1}{c}_1\otimes \dots \otimes c_n\otimes g+\sum _{i=1}^n{c}_1\otimes \dots \otimes c_{i-1}\otimes \Delta (c_i)\otimes c_{i+1}\otimes \dots \otimes c_n.$$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