Path algebras of quivers

introduction

Q quiver
a path in Q is a sequence $(i|\alpha_1,\alpha_2,\cdots,\alpha_l|j)$ such that $s(\alpha_i)=t(\alpha_{i-1})$ for all $i = 2, \cdots, l$
the path algebra of kQ of Q is the k-algebra with basis the set of all paths in Q with multiplication in the basis given by concatenation of two paths
path algebra of a quiver
- given a quiver Q, a path p is a sequence of arrows with some conditions
- path algebra : set of all k-linear combinations of all paths (including e_i's)
- p_1p_2 will correspond to a composition $p_2\circ p_1$ of two maps ($U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W$)