# nLab corepresentation

A left (right) corepresentation is a synonym for a left (right) coaction of a coalgebra (comonoid) in one of the monoidal categories which have linear, and possibly, functional connotation; e.g. in the context of topological vector spaces.

There is however also another meaning of a corepresentation for a Leibniz algebra.

Both a representation and a corepresentation of a right Leibniz $k$-algebra $\mathrm{\pi €}$ involve a $k$-module $M$ and two $k$-linear maps βactionsβ $M\beta \mathrm{\pi €}\beta M$ and $\mathrm{\pi €}\beta M\beta M$ with 3 axioms.

For representations:

$\left[m,\left[x,y\right]\right]=\left[\left[m,x\right],y\right]\beta \left[\left[m,y\right],x\right]$[m, [x, y]] = [[m, x], y] - [[m, y], x]
$\left[x,\left[a,y\right]\right]=\left[\left[x,m\right],y\right]\beta \left[\left[x,y\right],m\right]$[x, [a, y]] = [[x, m], y] - [[x, y], m]
$\left[x,\left[y,m\right]\right]=\left[\left[x,y\right],m\right]\beta \left[\left[x,m\right],y\right]$[x, [y, m]] = [[x, y], m] - [[x, m], y]

for $x,y\beta \mathrm{\pi €}$ and for $m\beta M$.

For corepresentatons:

$\left[\left[x,y\right],m\right]=\left[x,\left[y,m\right]\right]\beta \left[y,\left[x,m\right]\right]$[[x, y], m] = [x, [y, m]] - [y, [x, m]]
$\left[y,\left[a,x\right]\right]=\left[\left[y,m\right],x\right]\beta \left[m,\left[x,y\right]\right]$[y, [a, x]] = [[y, m], x] - [m, [x, y]]
$\left[\left[m,x\right],y\right]=\left[m,\left[x,y\right]\right]\beta \left[\left[y,m\right],x\right].$[[m, x], y] = [m, [x, y]] - [[y, m], x].

If the two βactionsβ are symmetric, i.e. $\left[x,m\right]+\left[m,x\right]=0$ for all $m\beta M$, $x\beta \mathrm{\pi €}$ then all the 6 axioms of representation or corepresentation are equivalent. If $M$ is underlying a Leibniz algebra then an action of $\mathrm{\pi €}$ on $M$ is by definition symmetric, hence all the 6 equivalent conditions hold.

Revised on November 23, 2012 15:21:03 by Zoran Ε koda (193.55.36.18)