Kupershmidt operators and related structures on Leibniz algebras

Kupershmidt operator is a key to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to Kupershmidt operators on Leibniz algebras and introduce (dual) KN-structures on a Leibniz algebra associated to a representation. It is proved that Kupershmidt operators and dual KN-structures can generate each other with certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual KN-structure. Finally, the notions of r-n structures,RBN-structures and BN-structures on Leibniz algebras are thoroughly studied and shown to bear interesting interrelations.Comment: 23 page

Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras

The goal of the present paper is to investigate non-abelian extensions of Lie-Yamaguti algebras and explore extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions in terms of Maurer-Cartan elements. Moreover, we discuss the equivalent conditions of the extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.Comment: arXiv admin note: text overlap with arXiv:2401.1487

Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras

In this paper, we investigate cohomologies and non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras. First, we consider Lie comodules and cohomologies of lambda-weighted Rota-Baxter Lie coalgebras. Next, we study non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras and classify the non-abelian extensions in terms of non-abelian cohomology group. Furthermore, we explore extensibility of a pair of automorphisms about a non-abelian extension of lambda-weighted Rota-Baxter Lie coalgebras, and derive the fundamental sequences of Wells in the context of lambda-weighted Rota-Baxter Lie coalgebras. Finally, we discuss the previous results in the case of abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras

Calderón-Zygmund operators and commutators on weighted Lorentz spaces

We find necessary conditions (which are also sufficient, for some particular cases) for a pair of weights u and w such that a Calder_on-Zygmund operator T, or its commutator [b; T], with b 2 BMO, is bounded on the weighted Lorentz spaces _p u(w), for 1 < p < 1. This result completes the study already known for the Hardy-Littlewood maximal operator and the Hilbert transform, and hence unifies the weighted theories for the Ap and Bp classes

Advance on the Application of Magnetic Field-assisted Freezing Technology in Food

Freezing is one of the most common and effective method of preserving food. However, the formation of large ice crystals during traditional freezing process will destroy food tissues and lead to quality deterioration. Therefore, how to improve the quality of frozen food by new freezing technology has become a research hotspot. Magnetic field-assisted freezing is a novel method for controlling ice crystal nucleation. The mechanism of magnetic field-regulated ice crystal nucleation and its applications in the fields of fruits and vegetables, livestock and poultry meat, cereals and other food products are reviewed in the present paper. According to the review results, although magnetic field freezing technology has been applied in many food fields, the current research mainly focuses on the effect of magnetic field on frozen food quality and freezing parameters, while there are few consensus on the mechanism of magnetic field-assisted freezing to regulate ice crystal nucleation. Therefore, more systematic research is required to reveal the mechanism of magnetic field-assisted freezing and promote the application of magnetic field-assisted freezing technology in the food field, to promote the quality of frozen food

Cohomologies of n-Lie Algebras with Derivations

The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation

A new approach to Hom-left-symmetric bialgebras

summary:The main purpose of this paper is to consider a new definition of Hom-left-symmetric bialgebra. The coboundary Hom-left-symmetric bialgebra is also studied. In particular, we give a necessary and sufficient condition that $s$-matrix is a solution of the Hom-$S$-equation by a cocycle condition