The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\operatorname{Hom}_H(H,\tau^{-n}H)$ where $\tau$ is the Auslander-Reiten translate.

As an $H$-module it is isomorphic to the direct sum of all the preprojective $H$-modules. The multiplication for this definition is given by the following:

if $\mu\colon H\to\tau^{-i}H$ and $\eta\colon H\to\tau^{-j}H$, then we take $\eta\mu$ to be the composition $\tau^{-i}\eta\mu$.

Here is one of the reasons (maybe the only reason) for the name. The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\operatorname{Hom}_H(H,\tau^{-n}H)$ where $\tau$ is the Auslander-Reiten translate.

As an $H$-module it is isomorphic to the direct sum of all the preprojective $H$-modules. The multiplication for this definition is given by the following:

if $\mu\colon H\to\tau^{-i}H$ and $\eta\colon H\to\tau^{-j}H$, then we take $\eta\mu$ to be the composition $(\tau^{-i}\eta)\mu$.

The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\Hom_H(H,\tau^{-n)H)$ where $\tau$ is the Auslander-Reiten translate.

As an $H$-module it is isomorphic to the direct sum of all the preprojective $H$-modules. The multiplication for this definition is given by the following:

if $\mu\colon H\to\tau^{-i}H$ and $\eta\colon H\to\tau^{-j}H$, then we take $\eta\mu$ to be the composition $\tau^{-i}\eta\mu$.

The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\operatorname{Hom}_H(H,\tau^{-n}H)$ where $\tau$ is the Auslander-Reiten translate.

As an $H$-module it is isomorphic to the direct sum of all the preprojective $H$-modules. The multiplication for this definition is given by the following:

if $\mu\colon H\to\tau^{-i}H$ and $\eta\colon H\to\tau^{-j}H$, then we take $\eta\mu$ to be the composition $\tau^{-i}\eta\mu$.