Stable rank of Leavitt path algebras

We characterize the values of the stable rank for Leavitt path algebras by giving concrete criteria in terms of properties of the underlying graph.

     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Simple flat Leavitt path algebras are von Neumann regular

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40?years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras.     

Graded irreducible representations of Leavitt path algebras: A new type and complete classification

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type.Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in Rangaswamy (2013) [15].     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Bases of Leavitt path algebras with only strongly regular elements

Recent articles consider invertible and locally invertible algebras (respectively, those having a basis consisting solely of invertible or solely of strongly regular elements). Previous contributions to the subject include the study of when Leavitt path algebras are invertible. This article investigates the local invertibility property in Leavitt path algebras. A complete classification of strongly regular monomials in Leavitt path algebras is given. Additionally, it is show that all directly finite and (von Neumann) regular Leavitt path algebras are locally invertible. It is also shown that a Leavitt path algebra has a basis consisting solely of strongly regular monomials if and only if it is commutative.     

On the Simplicity of the Lie Algebra of a Leavitt Path Algebra

For a field F and a row-finite directed graph Γ, let L(Γ) be the associated Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra [L(Γ), L(Γ)] to be simple.     

Simplicity of the Lie Algebra of Skew Symmetric Elements of a Leavitt Path Algebra

For a field F of characteristic not 2 and a directed row-finite graph Γ let L(Γ) be the Leavitt Path Algebra with standard involution *. We study the lie algebra of K = K(L(Γ), *) of * ?skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra [K, K] to be simple.     

The graded Grothendieck group and the classification of Leavitt path algebras

This paper is an attempt to show that, parallel to Elliott’s classification of AF C*-algebras by means of K-theory, the graded K ₀-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and multi-headed comets or rose graphs (graphs in which each head is connected to a cycle or to a collection of loops), or a mixture of these graphs (i.e., polycephaly graphs).     

Tilting modules over path algebras of Dynkin type and complete slice modules

Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different (?)-orbits of the AR-quiverof A,then T is a complete slice module.     

有限自动机的积与路代数的张量积

讨论了两个有限自动机积的路代数,得出了这些积的路代数与这两个有限自动机路代数的张量积的一些关系.     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

The Type Number of the Cosymplectic Hypersurfaces of 6-Dimensional Hermitian Submanifolds of the Cayley Algebra

We study the 6-dimensional oriented submanifolds of the Cayley algebra which are endowed with the Hermitian structure induced by 3-folds vector cross products. We prove that the type number of a cosymplectic hypersurface of a 6-dimensional Hermitian submanifold of the Cayley algebra is at most 3 and that a 6-dimensional Kaehler submanifold of the octave algebra has no cosymplectic hypersurfaces with the type number greater than one.     

Amalgamation property for the class of basic algebras and some of its natural subclasses

We study Basic algebra, the algebraic structure associated with basic propositional calculus, and some of its natural extensions. Among other things, we prove the amalgamation property for the class of Basic algebras, faithful Basic algebras and linear faithful Basic algebras. We also show that a faithful theory has the interpolation property if and only if its correspondence class of algebras has the amalgamation property.     

不可分素C^k—代数与本原C^＊—代数的讨论

本文证明：若Ａ是不可分的素Ｃ＾＊－代数，且包含非０的Ｌｉｍｉｎａｌ遗传Ｃ＾＊－子代数，则Ａ是本原Ｃ＾＊－代数，本文还给出了Ｉ型Ｃ＾＊－代数为本原Ｃ＾＊－代数的充要条件。     

Tychonoff's theorem for locally compact spaces and an elementary approach to the topology of path spaces

The path spaces of a directed graph play an important role in the study of graph -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.