arXiv subjects
Publications and source records attributed to Jocelyn P. Vilela.
We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted Steinberg algebras and that the associated graded discrete twist is unique. Our results include all discrete group algebras, and so are more general than the ungraded version.
In this paper, we give conditions on a unital non-associative ring $R$ and additive map $σ: R\longrightarrow R$ so that the non-associative skew power series ring, denoted by $R[[X;σ]]$ and the non-associative skew Laurent series ring denoted by $R((X;σ))$ are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of $R[[X;σ]]$ and $R((X;σ))$, respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for $R[[X;σ]]$ and $R((X;σ))$, we introduce and utilize the right (resp. left) quasi-associative property of $R$, which is equivalent to $rR$ (resp. $Rc$) being one-sided ideals for all $r\in R$ (resp. $c\in R$).
In this article, we prove an isomorphism theorem for the case of refinement $Γ$-monoids. Based on this we show a version of the well-known Jordan-Hölder theorem in this framework. The main theorem of this article states that - as in the case of modules - a monoid $T$ has a $Γ$-composition series if and only if it is both $Γ$-Noetherian and $Γ$-Artinian. As in module theory, these two concepts can be defined via ascending and descending chains respectively.
We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph $E$ consists of disjoint cycles precisely when its talented monoid $T_E$ has a certain Jordan-Hölder composition series. These are graphs whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K (E))$, we conclude that for graphs $E$ and $F$, if $K_0^{gr}(L_K (E))\cong K_0^{gr}(L_K (F))$ then $GKdim L_K(E) = GKdim L_K(F)$, thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.