Triple Derivations and Triple Homomorphisms of Perfect Lie Superalgebras

In this paper, we study triple derivations and triple homomorphisms of perfect Lie superalgebras over a commutative ring $R$. It is proved that, if the base ring contains $\frac{1}{2}$, $L$ is a perfect Lie superalgebra with zero center, then every triple derivation of $L$ is a derivation, and every triple derivation of the derivation algebra $Der (L)$ is an inner derivation. Let $L,~L^{'}$ be Lie superalgebras over a commutative ring $R$, the notion of triple homomorphism from $L$ to $L^{'}$ is introduced. We proved that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.Comment: 12pages in Indagationes Mathematicae, 201

Biphenyl Phytoalexin in Sorbus pohuashanensis Suspension Cell Induced by Yeast Extract

Biphenyls are unique phytoalexins de novo synthesized in plants in response to pathogen attack. These compounds are found in Maloideae, a subfamily of the Rosaceae. The anti-microbial activities of biphenyls have been reported in a number of studies and they appear to represent an important defense strategy against pathogens common in the Maloideae, such as species in Malus, Pyrus, Sorbus, and Chaenomeles. Here, cell suspension cultures of Sorbus pohuashanensis were established to study biphenyl phytoalexins formation after yeast extract (YE) treatment. An ultra-performance liquid chromatography (UPLC) method coupled with quadrupole time of flight mass spectrometry (Q-TOF-MS) LC−MS/MS was applied to determine the time course of these biphenyl biomarkers accumulation in YE-treated S. pohuashanensis suspension cells. The results of quantitative analyses show the content of Noraucuparin, 2′-Hydroxyaucuparin, and their glycosides initially increased, then decreased over time. The Noraucuparin content reached its highest (225.76 μg·g−1) at 18 h after treatment, 6 hours earlier than that of Noraucuparin 5-O-β-d-glucopyranoside. The content of 2′-Hydroxyaucuparin reached its highest (422.75 μg·g−1) at 30 h after treatment, also earlier than that of its glycoside. The understanding of phytoalexin metabolism in this study may provide a basis for improving Maloideae resistance to pathogens

L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal

Generalized derivations of Lie triple systems

In this paper, we present some basic properties concerning the derivation algebra Der (T), the quasiderivation algebra QDer (T) and the generalized derivation algebra GDer (T) of a Lie triple system T, with the relationship Der (T) ⊆ QDer (T) ⊆ GDer (T) ⊆ End (T). Furthermore, we completely determine those Lie triple systems T with condition QDer (T) = End (T). We also show that the quasiderivations of T can be embedded as derivations in a larger Lie triple system

Biderivations and linear commuting maps on the restricted contact lie algebras K(n; 1)

Let K be the restricted contact Lie algebras K(n; 1) over an algebraically closed field F of characteristic p > 3. We prove that each skew-symmetric biderivation of K is inner and show that commuting maps on K are scalar multiplication maps. Moreover, it is showed that the commuting automorphisms and dertivations of K are proved to be the identity mappings and zero mappings, respectively. Meanwhile, the commutative post-Lie algebra structures on K are given