Subalgebra depths within the path algebra of an acyclic quiver

TL;DR

This paper investigates the depth of specific subalgebras within the path algebra of an acyclic quiver, providing explicit calculations and bounds that are independent of the quiver's size.

Contribution

It computes the depth of the top subalgebra and the primary arrow subalgebra in acyclic quiver algebras, establishing their fixed depths regardless of the number of vertices.

Findings

Depth of top subalgebra is 3

Depth of primary arrow subalgebra is 4

Upper bounds on depth for subalgebra quotients

Abstract

Constraints are given on the depth of diagonal subalgebras in generalized triangular matrix algebras. The depth of the top subalgebra B = A /rad A in a finite, connected, acyclic quiver algebra A over an algebraically closed field K is then computed. Also the depth of the primary arrow subalgebra 1K + rad A = B in A is obtained. The two types of subalgebras have depths 3 and 4 respectively, independent of the number of vertices. An upper bound on depth is obtained for the quotient of a subalgebra pair.