This paper investigates the transposed Poisson structures on the Schrödinger algebra [Formula: see text] associated with [Formula: see text]-dimensional space-time of the Schrödinger Lie group. We prove that for [Formula: see text], the algebra [Formula: see text] admits no nontrivial [Formula: see text]-derivations and, consequently, no nontrivial transposed Poisson structures. In contrast, for the case [Formula: see text], we explicitly determine all [Formula: see text]-derivations and the corresponding transposed Poisson structures on [Formula: see text]. Additionally, we demonstrate that [Formula: see text] admits a nontrivial Hom-Lie structure.

In this article, we investigate the Green ring of the small quantum group [Formula: see text], where [Formula: see text] is a root of unity of order [Formula: see text] and [Formula: see text] is even. We describe the structures of the Green rings by generators with relations.

In this paper, we determine the unique graph whose least eigenvalue attains the minimum among all the connected simple graphs with given clique number.

For smooth projective varieties [Formula: see text] and [Formula: see text] over a perfect field [Formula: see text] of positive characteristic [Formula: see text] so that [Formula: see text] and [Formula: see text] are derived equivalent, and admitting a closed embedding into [Formula: see text], we assume that [Formula: see text]. We prove that if [Formula: see text] is ordinary (resp., Hodge-Witt), then [Formula: see text] is ordinary (resp., Hodge-Witt).

In order to induce the associative classical Yang-Baxter equations of any weight, we present the notion of an antisymmetric infinitesimal (ASI) bialgebra of weight [Formula: see text], extending the ASI bialgebras to any weight. Meanwhile, we consider the BiHom-deformation of the bialgebra above, which leads to the major research object that we need: nonhomogeneous associative BiHom-classical Yang-Baxter equations (abhcYBes). Subsequently, we focus on the characterizations and constructions of abhcYBes from generalized [Formula: see text]-operators and weighted Rota-Baxter operators, which can be seen as a generalization of the main results in [Adv. Theor. Math. Phys. 26 (2022) 1965–2009].

This paper is devoted to the structure of complete and perfect Jordan algebras, which are called sympathetic Jordan algebras here. In particular, every perfect Jordan algebra [Formula: see text] contains a greatest special sympathetic ideal [Formula: see text] and a decomposition into direct sum of ideals about the greatest special sympathetic ideal. Besides, each perfect Jordan algebra [Formula: see text] also contains a solvable ideal [Formula: see text] which is greatest among the solvable ideals [Formula: see text] of [Formula: see text] such that [Formula: see text] and a decomposition into direct sum of subalgebras [Formula: see text], where [Formula: see text] is a sympathetic subalgebra of [Formula: see text], which is similar to the Levi decomposition of Lie algebras. Moreover, [Formula: see text] is sympathetic if and only if [Formula: see text]. What is more, a class of ideals of Jordan algebras such that the quotients are sympathetic Jordan algebras are studied and some vital properties about this kind of ideal are highlighted.

Our purpose is to determine skew-symmetric biderivations [Formula: see text] and commuting linear maps [Formula: see text] on a multiplicative Hom-Lie algebra [Formula: see text] having their ranges in an [Formula: see text]-module [Formula: see text], which are both closely related to [Formula: see text], the centroid of [Formula: see text] on [Formula: see text]. We give the relationship between biderivations and commuting linear maps on a regular Hom-Lie algebra and those on the related Lie algebra. Under appropriate assumptions, we also prove that every [Formula: see text] in [Formula: see text] is of the form [Formula: see text] for some [Formula: see text], and [Formula: see text] coincides with [Formula: see text]. Besides, we give the algorithms for describing [Formula: see text] and [Formula: see text].

The ranks of free semimodules and semimodule types of a semiring [Formula: see text] are investigated. We show that a given semiring can admit only certain characteristic types of finitely generated free semimodules, and semirings are classified according to their semimodule types. In addition, we discuss the least cardinality of a finitely generated free [Formula: see text]-semimodule and prove that if [Formula: see text] has a semimodule of type [Formula: see text], then every [Formula: see text]-semimodule with free basis of cardinality [Formula: see text] ([Formula: see text]) has no generating set of cardinality less than [Formula: see text].

We first introduce the notion of an HLeibDer triple, which includes a Hom-Leibniz algebra and a derivation. Then we define a cohomology theory for HLeibDer triples with coefficients in a representation. Central extensions of an HLeibDer triple are also studied. Finally, we consider homotopy derivations on [Formula: see text] algebras and 2-derivations on Hom-Leibniz 2-algebras. The category of 2-term [Formula: see text] algebras with homotopy derivations is equivalent to the category of Hom-Leibniz 2-algebras with 2-derivations.

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.