Representations of Bihom-Lie Algebras

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

New Results on EP Elements in Rings with Involution

In this paper, we mainly give characterizations of EP elements in terms of equations. In addition, a related notion named a central EP element is defined and investigated. Finally, we focus on characterizations of a generalized EP element, i.e., [Formula: see text]-DMP element.

G-Algebra Structure on the Higher Order Hochschild Cohomology HS2∗(A,A)

We present a deformation theory associated to the higher Hochschild cohomology [Formula: see text]. We also study a [Formula: see text]-algebra structure associated to this deformation theory.

On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Level -1/2 Realization of Quantum N-Toroidal Algebra in Type Cn

We construct a level -1/2 vertex representation of the quantum [Formula: see text]-toroidal algebra of type [Formula: see text], which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type [Formula: see text] as a by-product.

Three Ideals of Lie Superalgebras

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.

On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

On Finite Local Rings with Clique Number Four

We study the algebraic structure of rings [Formula: see text] whose zero-divisor graph [Formula: see text]has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.

Open Frobenius Cluster-Tilted Algebras

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.