On finite-dimensional algebras isomorphic to monomial algebras.

• Main Author: Du, Jianbo.
• Other Authors: Burgess, Walter D.
• Format: Others
• Published: University of Ottawa (Canada) 2009
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/10393/10126; http://dx.doi.org/10.20381/ruor-16674

Let K be a field and $\Gamma$ = $(\Gamma\sb0,\Gamma\sb1)$ be a connected finite directed graph. Then $K\Gamma$ denotes the path algebra of $\Gamma$ over K. A two-sided ideal I of $K\Gamma$ is admissible in case there exists a positive integer $N \ge 2$ such that $\langle \Gamma\sb1\ \rangle\sp{N}\ \subset\ I\ \subset\ \langle \Gamma\sb1\ \rangle\sp2.$ It is well-known that every connected split basic finite dimensional K-algebra A is isomorphic to the quotient $K\Gamma/I$ of a path algebra $K\Gamma$ by an admissible ideal I (cf. (16)). (The underlying connected finite directed graph $\Gamma$ is usually called the quiver of A.) A quotient A = $K\Gamma/I$ of path algebra by admissible ideal is called a (finite dimensional) monomial algebra if I is generated by a set of finite directed paths in $\Gamma.$ In recent years the class of monomial algebras has been exposed to extensive study and significant advances have been made toward the understanding of the homological nature of this class of finite dimensional algebras (see (20), (21), (22) and (23)). We say that A = $K\Gamma/I$ is of monomial presentation type if A is isomorphic to a monomial algebra over K. The primary purpose of this thesis is to identify the path algebra quotients $K\Gamma$ of monomial presentation type under various conditions on the underlying finite directed graph $\Gamma$ and/or the quotients $K\Gamma/I$ themselves. Even though it is easy to determine whether or not a path algebra quotient is a monomial algebra, the identification of path algebra quotients of monomial presentation type is a fundamentally harder problem. Our study is of significance in that monomial algebras are known to be well behaved and many of their properties, such as homological ones, are algebra isomorphism invariants. The main tools to be utilized in the thesis are the noncommutative Grobner basis theory for path algebras and Saorin's change of variable method. In Chapter 2, we use these tools to find cases where path algebra quotients of monomial presentation type must already be monomial. We also study isomorphisms between monomial algebras and present a combinatorial criterion for two monomial algebras to be isomorphic. Chapter 3 is concerned with homological aspects of path algebra quotients of monomial presentation type. Specifically, we shall study minimality of the Anick-Green resolutions for the simple modules over path algebra quotients and discuss its relationship to the monomial presentation type. In the last chapter of the thesis, we devote ourselves to a particular class of quotients of path algebras--called right monomial rings--which were introduced by Burgess et al. in (6) as a generalization of monomial algebras. We shall answer an open question presented in (6) in some special cases, and provide a characterization of the monomial presentation type within the class of path algebra quotients which are right monomial rings by showing that they are precisely those gradable by the radical.