Path algebras of quivers
imported>Pythagoras0님의 2013년 10월 8일 (화) 12:01 판 (새 문서: ==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 algebr...)
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 (\(U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W\))