arXiv subjects

Ashish K. Srivastava

Publications and source records attributed to Ashish K. Srivastava.

At least 19 recordsLinked to original sources

arXiv2026

Quivers with quantum Yang-Baxter equation and Hecke condition: Deformation of face algebras

In this paper, we initiate the study of quivers carrying quantum Yang--Baxter and Hecke structure. Calling a quiver $Q$ to be a solution of the QYBE when the adjacency matrix of its Kronecker square is a solution, we show that this holds precisely when the adjacency matrix $A$ satisfies $A^2 = μA$ for a scalar $μ$. We determine exactly when Kulish's rank-one construction yields a Hecke $R$-matrix that satisfies both the braided QYBE and the Hecke condition. We deform Hayashi's face algebra by the resulting RTT relations, and prove that the quantum matrix algebra $\mathcal{O}_q(M_n)$ is isomorphic as a bialgebra to the Hecke-deformed face algebra of the rose quiver with $n$ petals, and that for a disjoint union of $m$ such roses the deformation is a genuine quantum groupoid with $m$-dimensional base, isomorphic as an algebra to $m^2$ copies of $\mathcal{O}_q(M_n)$.

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arXiv2026

Leavitt Path Algebra over Kronecker Square of Quivers and Cross product algebra

In this paper, we initiate the study of Leavitt path algebra over Kronecker square of a quiver and show the similarities and contrasts in the properties of Leavitt path algebra over a quiver and its Kronecker square. Furthermore, we discuss the connection of Leavitt path algebra over Kronecker square of a quiver with Hayashi's face algebra and the cross product algebra construction of the Leavitt path algebra over the original quiver.

math.RA↗

arXiv2024

Automorphisms of Leavitt path algebras: Zhang twist and irreducible representations

In this article, we construct (graded) automorphisms fixing all vertices of Leavitt path algebras of arbitrary graphs in terms of general linear groups over corners of these algebras. As an application, we study Zhang twist of Leavitt path algebras and describe new classes of irreducible representations of Leavitt path algebras of the rose graphs $R_n$ with $n$ petals.

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arXiv2022

MacWilliams extending conditions and quasi-Frobenius rings

MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929] and [F. M. Schneider, J. Zumbrägel, MacWilliams' extension theorem for infinite rings, Proc. Amer. Math. Soc. 147, 3 (2019), 947-961]. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if $R$ is a right (or left) artinian, left automorphism-invariant ring, then $R$ is quasi-Frobenius, thus answering a question asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929].

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arXiv2021

An Introduction to Supersymmetric Cluster Algebras

In this paper we propose the notion of cluster superalgebras which is a supersymmetric version of the classical cluster algebras introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a subalgebra of a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2|0; 4|1)$ of chiral conformal superspace (that is, $(2|0)$ planes inside the superspace $\mathbb C^{4|1}$) is a quotient of a cluster superalgebra.

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arXiv2021

Variations of primeness and Factorization of ideals in Leavitt Path Algebras

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra $L$ is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras $L$ in which every ideal admits such factorizations and also in which every ideal is one of these special type of ideals.

math.RA↗

arXiv2020

Endomorphism rings via minimal morphisms

We prove that if $u:K \rightarrow M$ is a left minimal extension, then there exists an isomorphism between two subrings, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of $K$ from those of the endomorphism ring of $M$ in certain situations such us when $K$ is invariant under endomorphisms of $M,$ or when $K$ is invariant under automorphisms of $M$.

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arXiv2020

Ziegler Partial Morphisms in additive exact categories

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

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arXiv2019

An introduction to a supersymmetric graph algebra

In this paper we propose a graph superalgebra which is the supersymmetric analogue of Leavitt path algebras. We find a basis for these superalgebras and characterize when they have polynomial growth.

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arXiv2018

Leavitt path algebras with bounded index of nilpotence

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L_{K}(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an exit and there is a fixed positive integer $n$ such that the number of distinct paths that end at any given vertex $v$ (including $v$, but not including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal to $n$. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial identity. Furthermore, Leavitt path algebras with bounded index of nilpotence are shown to be directly-finite and to be $\mathbb{Z}$-graded $Σ$-$V$ rings. As an application of our results, we answer an open question raised in \cite{JST} whether an exchange $Σ$-$V$ ring has bounded index of nilpotence.

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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↗

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

The Schröder-Bernstein problem for Modules

In this paper we study the Schröder-Bernstein problem for modules. We obtain a positive solution for the Schröder-Bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied, for example, in the case of injective, pure-injective or cotorsion envelopes. In the particular cases of injective envelopes and pure-injective envelopes, we are able to extend it further and we show that the Schröder-Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes.

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arXiv2016

Additive unit structure of endomorphism rings and invariance of modules

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples that show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case.

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arXiv2016

Leavitt path algebras with bounded index of nilpotence and simple modules over them

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence and show that each graded simple module $S$ over a Leavitt path algebra with bounded index of nilpotence is graded $Σ$-injective, that is, $S^{(α)}$ is graded injective for any cardinal $α$. Furthermore, we characterize Leavitt path algebras over which each simple module is $Σ$-injective. We have shown that each simple module over a Leavitt path algebra $L_K(E)$ is $Σ$-injective if and only if the graph $E$ contains no cycles, and there is a positive integer $d$ such that the length of any path in $E$ is less than or equal to $d$ and the number of distinct paths ending at any vertex $v$ (including $v$) is less than or equal to $d$.

math.RA↗

arXiv2016

Modules which are coinvariant under automorphisms of their projective covers

In this paper we study modules coinvariant under automorphisms of their projective covers. We first provide an alternative, and in fact, a more succinct and conceptual proof for the result that a module $M$ is invariant under automorphisms of its injective envelope if and only if given any submodule $N$ of $M$, any monomorphism $f:N\rightarrow M$ can be extended to an endomorphism of $M$ and then, as a dual of it, we show that over a right perfect ring, a module $M$ is coinvariant under automorphisms of its projective cover if and only if for every submodule $N$ of $M$, any epimorphism $φ: M\rightarrow M/N$ can be lifted to an endomorphism of $M$.

math.RA↗

arXiv2016

$V$-rings versus $Σ$-$V$ Rings

This paper studies similarities and differences between the classes of rings over which each simple module is injective and rings over which each simple module is $Σ$-injective. The rings in the former class are called $V$-rings and the rings in the latter class are called $Σ$-$V$ rings. We have obtained analogues of various well-known results about $V$-rings for $Σ$-$V$ rings. Motivated by a conjecture of Kaplansky, Fisher asked if a prime right $V$-ring is right primitive. Although a counter-example to Kaplansky's conjecture was constructed long ago but Fisher's question is still open. In this paper we show that for a right $Σ$-$V$ ring, the notions of prime and primitive are equivalent. Also, we show that an exchange $Σ$-$V$ ring is left-right symmetric and moreover, it is von Neumann regular.

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arXiv2015

Rings with each right ideal automorphism-invariant

In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indecomposable right artinian right non-singular right $a$-ring as a triangular matrix ring of certain block matrices.

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