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 $\mathrm{\pi \x9d\x94\u20ac}$ involve a $k$-module $M$ and two $k$-linear maps βactionsβ $M\beta \x8a\x97\mathrm{\pi \x9d\x94\u20ac}\beta \x86\x92M$ and $\mathrm{\pi \x9d\x94\u20ac}\beta \x8a\x97M\beta \x86\x92M$ with 3 axioms.

For representations:

$$[m,[x,y]]=[[m,x],y]\beta \x88\x92[[m,y],x]$$

$$[x,[a,y]]=[[x,m],y]\beta \x88\x92[[x,y],m]$$

$$[x,[y,m]]=[[x,y],m]\beta \x88\x92[[x,m],y]$$

for $x,y\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$ and for $m\beta \x88\x88M$.

For corepresentatons:

$$[[x,y],m]=[x,[y,m]]\beta \x88\x92[y,[x,m]]$$

$$[y,[a,x]]=[[y,m],x]\beta \x88\x92[m,[x,y]]$$

$$[[m,x],y]=[m,[x,y]]\beta \x88\x92[[y,m],x].$$

If the two βactionsβ are symmetric, i.e. $[x,m]+[m,x]=0$ for all $m\beta \x88\x88M$, $x\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$ then all the 6 axioms of representation or corepresentation are equivalent. If $M$ is underlying a Leibniz algebra then an action of $\mathrm{\pi \x9d\x94\u20ac}$ 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)