# admissible ideals,, bound quiver and its algebra

Assume, that $Q$ is a quiver and $k$ is a field. Let $kQ$ be the associated path algebra. Denote by ${R}_{Q}$ the two-sided ideal^{} in $kQ$ generated by all paths of length $1$, i.e. all arrows. This ideal is known as the arrow ideal.

It is easy to see, that for any $m\u2a7e1$ we have that ${R}_{Q}^{m}$ is a two-sided ideal generated by all paths of length $m$. Note, that we have the following chain of ideals:

$${R}_{Q}^{2}\supseteq {R}_{Q}^{3}\supseteq {R}_{Q}^{4}\supseteq \mathrm{\cdots}$$ |

Definition. A two-sided ideal $I$ in $kQ$ is said to be admissible if there exists $m\u2a7e2$ such that

$${R}_{Q}^{m}\subseteq I\subseteq {R}_{Q}^{2}.$$ |

If $I$ is an admissible ideal in $kQ$, then the pair $(Q,I)$ is said to be a bound quiver and the quotient algebra $kQ/I$ is called bound quiver algebra.

The idea behind this is to treat some paths in a quiver as equivalent^{}. For example consider the following quiver

$$\text{xymatrix}\mathrm{\&}2\text{ar}{[dr]}^{b}\mathrm{\&}1\text{ar}{[rr]}^{c}\text{ar}{[dr]}_{e}\text{ar}{[ur]}^{a}\mathrm{\&}\mathrm{\&}3\mathrm{\&}4\text{ar}{[ur]}_{f}$$ |

Then the ideal generated by $ab-c$ is not admissible ($ab-c\notin {R}_{Q}^{2}$) but an ideal generated by $ab-ef$ is. We can see that this means that ,,walking” from $1$ to $3$ directly and through $2$ is not the same, but walking in the same number of steps is.

Note, that in our case there is no path of length greater then $2$. In particular, for any $m>2$ we have ${R}_{Q}^{m}=0$.

More generally, it can be easily checked, that if $Q$ is a finite quiver without oriented cycles, then there exists $m\in \mathbb{N}$ such that ${R}_{Q}^{m}=0$

Title | admissible ideals,, bound quiver and its algebra |
---|---|

Canonical name | AdmissibleIdealsBoundQuiverAndItsAlgebra |

Date of creation | 2013-03-22 19:16:42 |

Last modified on | 2013-03-22 19:16:42 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 6 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |