Narrow Results

This paper determines all local and 2-local derivations of the 3-dimensional complex simple Lie algebra $𝔰𝔩(2)$ on its any finite-dimensional completely reducible module. In particular, the quotient space of the one consisting of all local derivations by the one consisting of all derivations of $𝔰𝔩(2)$ on its $(n+1)$-dimensional simple module has the dimension $n-1$ if $n$ is odd, and $0$ otherwise. On the other hand, each 2-local derivation of $𝔰𝔩(2)$ on any finite-dimensional completely reducible module is a derivation.

In this paper, we investigate the algebra $U_q^{\mathrm{\Delta }}({𝔰𝔩}_2)$, which is known as the equitable $q$-deformation of ${𝔰𝔩}_2$. This algebra was introduced in 2005 by Ito et al. Assume that $q$ is a primitive $m$th root of unity with $m\ge 3$. We prove that $U_q^{\mathrm{\Delta }}({𝔰𝔩}_2)$ becomes a Polynomial Identity (PI) algebra. It was previously known for such algebras, the simple modules are finite-dimensional with dimension at most the PI degree. We determine the PI degree of $U_q^{\mathrm{\Delta }}({𝔰𝔩}_2)$, and we classify up to isomorphism the simple $U_q^{\mathrm{\Delta }}({𝔰𝔩}_2)$-modules. We also find the center of $U_q^{\mathrm{\Delta }}({𝔰𝔩}_2)$.

In this paper, we consider a class of non-weight modules for some algebras related to the Virasoro algebra: The algebra Vir(a, b), the twisted deformative Schrödinger–Virasoro Lie algebras and the Schrödinger algebra. We study the modules whose restriction to the Cartan subalgebra (modulo center) are free of rank 1 for these algebras. Moreover, the simplicities of these modules are determined.

Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

Simple restricted modules are considered for the restricted contact Lie superalgebras of odd type over an algebraically closed field with characteristic $p>2$. In particular, a sufficient and necessary condition in terms of typical or atypical weights is given for the restricted Kac modules to be simple. Furthermore, the number of the simple restricted Kac modules is obtained.

Let $𝔤$ be the special linear Lie algebra ${𝔰𝔩}_3$ of rank 2 over an algebraically closed field $k$ of characteristic 3. In this paper, we classify all irreducible representations of $𝔤$, which completes the classification of the irreducible representations of ${𝔰𝔩}_3$ over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants of $𝔤$.

In this paper, we construct and study some non-weight modules for the Heisenberg–Virasoro algebra and the $W$ algebra $W(2,2)$. We determine the modules, whose restriction to the universal enveloping algebra of the degree-$0$ part (modulo center) are free of rank $1$ for these two algebras. In the most interesting case, this degree-$0$ part is not the Cartan subalgebra. We also determine the simplicity of these modules, which provide new simple modules for the $W$ algebra $W(2,2)$.

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 $\mathrm{\Sigma }$–$V$ rings. As an application of our results, we answer an open question raised in [S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs (Oxford University Press, 2012)] whether an exchange $\mathrm{\Sigma }$–$V$ ring has bounded index of nilpotence.

Let $A$ be an Artin algebra. It is well known that $A$ is selfinjective if and only if every finitely generated $A$-module is reflexive. In this paper, we pose and motivate the question whether an algebra $A$ is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.

In this paper, we explore the possibility of endowing simple infinite-dimensional ${𝔰𝔩}_2(ℂ)$-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in [7,8].

Over a field of characteristic $p>0$, the first cohomology of the orthogonal symplectic Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1,2)$ with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1,2)$. As a byproduct, the first cohomology of $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1,2)$ with coefficients in the restricted enveloping algebra (under the adjoint action) is not trivial.

The modules which are isomorphic to their non-zero submodules are known as iso-retractable. We characterize simple modules in terms of iso-retractable modules. We provide several sufficient conditions for iso-retractable modules to be simple. We show that if the endomorphism ring of an iso-retractable module is von-Neumann regular then $M$ is a simple module. In general, iso-retractable modules need not be projective (injective) and vice versa. We investigate some properties of iso-retractable modules with projectivity as well as injectivity. Finally, we provide some open problems.