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 kk-algebra 𝔀\mathfrak{g} involve a kk-module MM and two kk-linear maps β€œactions” MβŠ—π”€β†’MM\otimes\mathfrak{g}\to M and π”€βŠ—Mβ†’M\mathfrak{g}\otimes M\to M with 3 axioms.

For representations:

[m,[x,y]]=[[m,x],y]βˆ’[[m,y],x][m, [x, y]] = [[m, x], y] - [[m, y], x]
[x,[a,y]]=[[x,m],y]βˆ’[[x,y],m][x, [a, y]] = [[x, m], y] - [[x, y], m]
[x,[y,m]]=[[x,y],m]βˆ’[[x,m],y][x, [y, m]] = [[x, y], m] - [[x, m], y]

for x,yβˆˆπ”€x,y\in\mathfrak{g} and for m∈Mm\in M.

For corepresentatons:

[[x,y],m]=[x,[y,m]]βˆ’[y,[x,m]][[x, y], m] = [x, [y, m]] - [y, [x, m]]
[y,[a,x]]=[[y,m],x]βˆ’[m,[x,y]][y, [a, x]] = [[y, m], x] - [m, [x, y]]
[[m,x],y]=[m,[x,y]]βˆ’[[y,m],x].[[m, x], y] = [m, [x, y]] - [[y, m], x].

If the two β€œactions” are symmetric, i.e. [x,m]+[m,x]=0[x,m] + [m,x] = 0 for all m∈Mm\in M, xβˆˆπ”€x\in\mathfrak{g} then all the 6 axioms of representation or corepresentation are equivalent. If MM is underlying a Leibniz algebra then an action of 𝔀\mathfrak{g} on MM is by definition symmetric, hence all the 6 equivalent conditions hold.

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