arXiv subjects

R. Hazrat

Publications and source records attributed to R. Hazrat.

At least 19 recordsLinked to original sources

arXiv2023

Williams' Conjecture holds for meteor graphs

A meteor graph is a connected graph with no sources and sinks consisting of two disjoint cycles and the paths connecting these cycles. We prove that two meteor graphs are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K_0^{gr}$, are $\mathbb Z [x,x^{-1}]$-module isomorphic. As a consequence, the Leavitt path algebras of meteor graphs are graded Morita equivalent if and only if their graph $C^*$-algebras are equivariant Morita equivalent.

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arXiv2020

Realizing ultragraph Leavitt path algebras as Steinberg algebras

In this article, we realize ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these algebras. Using skew product groupoid, we show that ultragraph Leavitt path algebras are graded von Neumann regular rings. We characterize strongly graded ultragraph Leavitt path algebras and show that every ultragraph Leavitt path algebra is semiprimitive. Moreover, we characterize irreducible representations of ultragraph Leavitt path algebras. We also show that ultragraph Leavitt path algebras can be realized as Cuntz-Pimsner rings.

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arXiv2015

The commutators of classical groups

In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 group. Namely, he proved that for an associative ring A with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems and methods developed include some of the splendid results of the heroes of classical algebraic K-theory; Bak, Quillen, Milnor, Suslin, Swan and Vaserstein, among others. One of the dominant techniques in establishing commutator type results is localisation. In this note we describe some recent applications of localisation methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups, GL(n,A), unitary groups GU(2n,A, Lambda) and Chevalley groups G(Phi,A). We also state some of the intermediate results as well as some corollaries of these results. This note provides a general overview of the subject and covers the current activities. It contains complete proofs of several main results to give the reader a self-contained source. We have borrowed some of the proofs from our previous papers and expositions

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arXiv2013

Multiplicative groups of division rings

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.

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arXiv2012

The dynamics of Leavitt path algebras

Recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the Morita theory and the Grothendieck group in the theory of Leavitt path algebras \cite{flowa}. In this paper we show that the notion of conjugacy of shifts of finite type is closely related to the {\it graded} Morita theory and consequently the {\it graded} Grothendieck group. This fits into the general framework we have in these two theories: Conjugacy yields the flow equivalence and the graded Morita equivalence can be lifted to the Morita equivalence. Starting from a finite directed graph, the observation that the graded Grothendieck group of the Leavitt path algebra associated to $E$ coincides with the Krieger dimension group of the shift of finite type associated to $E$ provides a link between the theory of Leavitt path algebras and symbolic dynamics. It has been conjectured that the ordered graded Grothendieck group as $\mathbb Z[x,x^{-1}]$-module (we call this the graded dimension group) classifies the Leavitt path algebras completely \cite{hazann}. Via the above correspondence, utilising the results from symbolic dynamics, we prove that for two purely infinite simple unital Leavitt path algebras, if their graded dimension groups are isomorphic, then the algebras are isomorphic.

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arXiv2012

A note on the isomorphism conjectures for Leavitt path algebras

We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K_0 classifies them completely. For arbitrary Leavitt path algebras, it is conjectured that K^{\gr}_0 classifies them completely \cite{hazann}. We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple ones) if their K^{\gr}_0-groups of their Leavitt path algebras are isomorphic then their K_0-groups are isomorphic as well.

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arXiv2011

The graded structure of Leavitt Path algebras

Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\mathbb Z$-grading and characterize the ($\mathbb Z$-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, $C_n$-comet and multi-headed graphs. The last two type are examples of graphs whose Leavitt path algebras are strongly graded. We characterize Leavitt path algebras which are strongly graded, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural $\textstyle{\bigoplus} \mathbb Z$-grading and in their simplest form recover the Leavitt algebras $L(n,k)$. We then establish some basic properties of these algebras.

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arXiv2011

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 comet and polycephaly graphs (graphs which each head is connected to a cycle or a collection of loops).

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arXiv2011

Homogeneous SK1 of simple graded algebras

For a simple graded algebra A=M_n(E) over a graded division algebra E, a short exact sequence relating the reduced Whitehead group of the homogeneous part of A to that of E is established. In particular it is shown that the homogeneous SK1 is not in general Morita invariant.

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arXiv2011

Multiple Commutator Formulas

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \GL_n(A,I_i) the principal congruence subgroup of level I_i in GL_n(A) and E_n(A,I_i) be the relative elementary subgroup of level I_i. We prove a multiple commutator formula [E_n(A,I_0),\GL_n(A,I_1),& \GL_n(A, I_2),..., \GL_n(A, I_m)] = [E_n(A,I_0),E_n(A,I_1),E_n(A, I_2),..., E_n(A, I_m)], which is a broad generalization of the standard commutator formulas.

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arXiv2011

K-theory of Azumaya algebras over schemes

Let $X$ be a connected, noetherian scheme and $\mathcal{A}$ be a sheaf of Azumaya algebras on $X$ which is a locally free $\mathcal{O}_{X}$-module of rank $a$. We show that the kernel and cokernel of $K_{i}(X) \to K_{i}(\mathcal{A}) $ are torsion groups with exponent $a^{m}$ for some $m$ and any $i\geq 0$, when $X$ is regular or $X$ is of dimension $d$ with an ample sheaf (in this case $m\leq d+1$). As a consequence, $K_{i}(X,\mathbb Z/m)\cong K_{i}(\mA,\mathbb Z/m)$, for any $m$ relatively prime to $a$.

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arXiv2011

On Quillen's calculation of graded $K$-theory

We adapt Quillen's calculation of graded K-groups of Z-graded rings with support in N to graded K-theory, allowing gradings in a product Z \times G with G an arbitrary group. This in turn allows us to use inductions and calculate graded K-theory of Z^m-graded rings. Here Z is the ring of integers and N positive natural numbers.

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arXiv2011

Relative unitary commutator calculus and applications

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, for two form ideals of a form ring. This answers two problems posed in a paper by Alexei Stepanov and the second author.

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arXiv2011

The yoga of commutators

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and enhanced versions of localisation-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localisation methods can be much more efficient, than expected.

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arXiv2010

On Graded Simple Algebras

This note begins by observing that a graded central simple algebra, graded by an abelian group, is a graded Azumaya algebra and it is free over its centre. For a graded Azumaya algebra A free over its centre $R$, we show that K_i^{\gr} (A) is "very close" to K_i^{\gr}(R), where K_i^{\gr} (R) is defined to be K_i(\Pgr (R)). Here \Pgr (R) is the category of graded finitely generated projective R-modules and K_i, \,i\geq 0, are the Quillen K-groups.

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arXiv2010

Iterated Leavitt Path Algebras

Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras L(1,n). In this note, we introduce iterated Leavitt path algebras associated to directed weighted graphs which have natural $\oplus \mathbb Z$ grading and in their simplest form recover the Leavitt algebras $L(n,k)$. We also characterize Leavitt path algebras which are strongly graded.

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arXiv2009

Unitary SK1 of graded and valued division algebras, I

The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.

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arXiv2008

SK1 of Azumaya algebras over hensel pairs

Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).

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