It is shown that an algebra $Λ$ can be lifted with nilpotent Jacobson radical $r = r(Λ)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with each $\bar e_{ii} $ in the center of $\bar Λ= Λ/r$ iff $Λ$ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, $Λ$ is a finite algebra with non-zero unity element over perfect field $k$ (e.g. a field with characteristic zero or a finite field) iff $Λ$ is isomorphic to a generalized path algebra $k (D, Ω, ρ)$ of finite directed graph with weak relations and $dim {\} Ω< \infty $; $Λ$ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff $Λ$ is isomorphic to a path algebra with relations.
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We also show that upper bounds coincide with the corresponding generalized quantum upper cluster algebras under the "coprimality" condition.
The so called generalized down-up algebras are revisited from a viewpoint of Gröbner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $λω\ne 0$), and by means of homogeneous Gröbner defining relations, the associated graded structures of generalized down-up algebras, namely the associated graded algebras, Rees algebras, and the homogenized algebras of generalized down-up algebras, are explored comprehensively.
The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand-Kirillov dimension and the entropy. We give a complete classification of path algebras over finite graphs by dimension, Gelfand-Kirillov dimension and algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic …
The purpose of this article is to show a close relationship between the generalized central series of Leibniz algebras. Some analogues of the classical group-theoretical theorems of Schur and Baer for Leibniz algebras are proved.
Weighted Leavitt path algebras were introduced in 2013 by Roozbeh Hazrat. These algebras generalise simultaneously the usual Leavitt path algebras and William Leavitt's algebras $L(m,n)$. In this paper we try to give an overview of what is known about the weighted Leavitt path algebras. We also prove some new results (in particular on the graded K-theory of weighted Leavitt path algebras) and mention open problems.
In this text, we are concerned with ring epimorphisms, and more specifically universal localisations, from path algebras to matrix algebras. We are mainly focused on constructing ring epimorphisms and universal localisations by extending them from from smaller path algebras to larger path algebras. At first, we discuss some simple results, and then we present generalizations thereof in several directions.
We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumpti…
We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path…
