arXiv subjects

Roozbeh Hazrat

Publications and source records attributed to Roozbeh Hazrat.

53 records · Page 3Linked to original sources

arXiv2018

Strongly graded groupoids and strongly graded Steinberg algebras

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $κ: \mathcal G \to Γ$, to a discrete group $Γ$, such that $κ^{-1}(γ)κ^{-1}(δ) = κ^{-1}(γδ)$, for all $γ, δ\in Γ$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_ε$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.

math.RA↗

arXiv2018

Structure theory of graded regular graded self-injective rings and applications

In this paper, we develop structure theory for graded regular graded self-injective rings and apply it in the context of Leavitt path algebras. We show that for a finite graph, graded regular graded self-injective Leavitt path algebras are of graded type I and these are precisely graded $Σ$-$V$ Leavitt path algebras.

math.RA↗

arXiv2018

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 a 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.

math.RA↗

arXiv2017

Leavitt path algebras: Graded direct-finiteness and graded $Σ$-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded $Σ$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $Σ$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $Σ$-$V$ ring $L$% . When the graph $E$ is finite, we show that $L$ is a graded $Σ$-$V$ ring $\Longleftrightarrow L$ is graded directly-finite $\Longleftrightarrow L $ has bounded index of nilpotence $\Longleftrightarrow $ $L$ is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph $E$ is infinite. Following this, we also characterize Leavitt path algebras $L$ which are non-graded $Σ$-$V$ rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra $L$ is a graded $Σ$-$V$ ring, then $L$ is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vaš \cite{V} on directly-finite Leavitt path algebras.

math.RA↗

arXiv2017

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.

math.RA↗

arXiv2017

A note on multiplicative commutators of division rings

We give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its centre, thus disproving the conjecture posed in the paper "Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8".

math.RA↗

arXiv2017

Graded Steinberg algebras and their representations

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite $k$-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.

math.KT↗

arXiv2017

Applications of normal forms for weighted Leavitt path algebras: simple rings and domains

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras $L_K(n, n + k)$ constructed by Leavitt. Using Bergman's Diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.

math.RA↗

arXiv2016

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.

math.RA↗

arXiv2016

Recollements, sinks elimination and Leavitt path algebras

For Leavitt path algebras, we show that whereas removing sources from a graph produces a Morita equivalence, removing sinks gives rise to a recollement situation. In general, we show that for a graph $E$ and a finite hereditary subset $H$ of $E^0$ there is a recollement $$\xymatrix{ L_K(E/\overline H) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E_H) \rModd .}$$ We record several corollaries.

math.RA↗

arXiv2016

Graded Rings and Graded Grothendieck Groups

This monograph is devoted to a comprehensive study of graded rings and graded K-theory. A bird's eye view of the graded module theory over a graded ring gives an impression of the module theory with the added adjective "graded" to all its statements. Once the grading is considered to be trivial, the graded theory reduces to the usual module theory. So from this perspective, the graded module theory can be considered as an extension of the module theory. However, one aspect that could be easily missed from such a panoramic view is that, the graded module theory comes equipped with a shifting, thanks to being able to partition the structures and rearranging these partitions. This adds an extra layer of structure (and complexity) to the theory. An sparkling example of this is the theory of graded Grothendieck groups, K^{gr}_0, which is the main focus of this monograph. Whereas the usual K_0 is an abelian group, thanks to the shiftings, K^{gr}_0 has a natural Z[Γ]-module structure, where Γis the graded group. As we will see throughout this note, this extra structure carries a substantial information about the graded ring.

math.RA↗

arXiv2015

On graded irreducible representations of Leavitt path algebras

Using the E-algebraic systems, various graded irreducible representations of a Leavitt path algebra L of a graph E over a field K are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by Laurent vertices and/or line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V_[p] induced by infinite path tail-equivalent to an infinite path p is graded if and only if p is an irrational infinite path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing a theorem of one of the co-authors that every Leavitt path algebra is graded von Neumann regular, we describe the graded self-injective Leavitt path algebras. These turn out to be direct sums of matrix rings of arbitrary size over K and k[x,x^-1].

math.RA↗

arXiv2013

Leavitt path algebras are graded von Neumann regular rings

In sharp contrast to the Abrams-Rangaswamy Theorem that the only von Neumann regular Leavitt path algebras are exactly those associated to acyclic graphs, here we prove that the Leavitt path algebra of any arbitrary graph is a graded von Neumann regular ring. Several properties of Leavitt path algebras, such as triviality of the Jacobson radical, flatness of graded modules and finitely generated graded right (left) ideals being generated by an idempotent element, follow as a consequence of general theory of grade von Neumann regular rings.

math.RA↗

arXiv2012

Generation of relative commutator subgroups in Chevalley groups

Let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. In the present paper we describe generators of the commutator groups of relative elementary subgroups $\big[E(Φ,R,I),E(Φ,R,J)\big]$ both as normal subgroups of the elementary Chevalley group $E(Φ,R)$, and as groups. Namely, let $x_{\a}(ξ)$, $\a\inΦ$, $ξ\in R$, be an elementary generator of $E(Φ,R)$. As a normal subgroup of the absolute elementary group $E(Φ,R)$, the relative elementary subgroup is generated by $x_{\a}(ξ)$, $\a\inΦ$, $ξ\in I$. Classical results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as a group $E(Φ,R,I)$ is generated by $z_{\a}(ξ,η)$, where $\a\inΦ$, $ξ\in I$, $η\in R$. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of $E(Φ,R)$ the relative commutator subgroup $\big[E(Φ,R,I),E(Φ,R,J)\big]$ is generated by the following three types of generators: i) $\big[x_α(ξ),z_α(ζ,η)\big]$, ii) $\big[x_α(ξ),x_{-α}(ζ)\big]$, and iii) $x_α(ξζ)$, where $α\inΦ$, $ξ\in I$, $ζ\in J$, $η\in R$. As a group, the generators are essentially the same, only that type iii) should be enlarged to iv) $z_α(ξζ,η)$. For classical groups, these results, with much more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results, namely in the recent work of Alexei Stepanov on relative commutator width.

math.RA↗

arXiv2012

Relative commutator calculus in Chevalley groups

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups $G(Φ,R)$, $\rk(Φ)\geq 2$, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in \cite{RZ,RNZ}. As an application we prove the mixed commutator formula, \[ \big [E(Φ,R,\ma),C(Φ,R,\mb)\big ]=\big [E(Φ,R,\ma),E(Φ,R,\mb)\big], \] for two ideals $\ma,\mb\unlhd R$. This answers a problem posed in a paper by Alexei Stepanov and the second author.

math.RA↗

arXiv2012

Commutator width in Chevalley groups

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

math.RA↗

arXiv2012

Multiple Commutator Formulas for Unitary Groups

Let $(\FormR)$ be a form ring such that $A$ is quasi-finite $R$-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups $\GU(2n,\FormR)$, $n\ge 3$. For a form ideal $(I,Γ)$ of the form ring $(\FormR)$ we denote by $\EU(2n,I,Γ)$ and $\GU(2n,I,Γ)$ the relative elementary group and the principal congruence subgroup of level $(I,Γ)$, respectively. Now, let $(I_i,Γ_i) $, $i=0,...,m$, be form ideals of the form ring $(A,Λ)$. The main result of the present paper is the following multiple commutator formula [\big[\EU(2n,I_0,Γ_0),&\GU(2n,I_1,Γ_1),\GU(2n, I_2,Γ_2),..., \GU(2n,I_m,Γ_m)\big]= &\big[\EU(2n,I_0,Γ_0),\EU(2n,I_1,Γ_1),\EU(2n,I_2,Γ_2),..., \EU(2n, I_m, Γ_m)\big],] which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classical like-groups over commutative and finite-dimensional rings.

math.RA↗