Relation between the regular representation and the group algebra.

Question

Consider a group $G$. If I have understood the concept correctly, the vector space $$ V=\operatorname{span}\{(\mathbf{e}_g)_{g\in G}\} $$ is the regular representation when we define the action of the group by $$ h\mathbf{e}_g=\mathbf{e}_{hg}. $$ If we replace $h$ by $\mathbf{e}_h$ in the previous equation, then we obtain the group algebra. Is this correct?

Answer 1

1. A regular representation of a group is a representation of the group $G$ on a vector space whose elements can be associated with the elements of the group. In the case of a left regular representation, the group action is defined as left multiplication: for a group element $h$ and a space element $g$ (which is associated with a group element), $h\mathbf{e}g=\mathbf{e}{hg} $.

2. A group algebra is a structure that can be built from a group by adding a multiplication operation. In the context of this question, the group algebra is obtained from the left regular representation by replacing the element $h$ of the group with the corresponding element $\mathbf{e}_h$ of the space. Thus the action of a group is defined in terms of the elements of the space, not the elements of the group.

Answer: Yes, that is correct. A group algebra is a left regular representation with an extra structure on it (multiplication that turns it into an algebra)​