Module and Comodule Structures on Path Space

On path space kQ, there is a trivial kQ-module structure determined by the multiplication of path algebra kQ and a trivial kQ-comodule structure determined by the comultiplication of path coalgebra kQ. In this paper, on path space kQ, a nontrivial kQ-module structure is defined, and it is proved that this nontrivial left kQ-module structure is isomorphic to the dual module structure of trivial right kQ-comodule. Dually, on path space kQ, a nontrivial kQ-comodule structure is defined, and it is proved that this nontrivial right kQ-comodule structure is isomorphic to the dual comodule structure of trivial left kQ-module. Finally, the trivial and nontrivial module structures on path space are compared from the aspect of submodule, and the trivial and nontrivial comodule structures on path space are compared from the aspect of subcomodule. Keywords—Quiver, path space, module, comodule, dual.