In this paper, all symmetric super-biderivations of some Lie superalgebras are determined. As an application, commutative post-Lie superalgebra structures on these Lie superalgebras are also obtained.

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455-2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest |G|? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt.

We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group. There is a quiver Q_G with relations ρ_G such that the skew group algebras A[G] is Morita equivalent to the quotient algebra of path algebra kQ_G modulo ideal (ρ_G). Generally, the quiver Q_G is not connected. In this paper we develop a method to determine the number of connect components of Q_G. Meanwhile, we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.

Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by $\mathfrak{B}_1$ and $\mathfrak{B}_2$, respectively. In this paper, a characterisation of groups in $\mathfrak{B}_1$ or in $\mathfrak{B}_2$ is given. By applying this new characterisation, some new properties of $\mathfrak{B}_1$ ($\mathfrak{B}_2$) and new proofs of many known results about $\mathfrak{B}_1$ or $\mathfrak{B}_2$ are obtained. Further, closure properties of $\mathfrak{B}_1$ and $\mathfrak{B}_2$ are discussed.

Suppose the ground field F is an algebraically closed field characteristic of p>2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra L^λp,p structures parameterized by elements λ∈F^p. We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of L^λp,p with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of L^λp,p with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let $G$ be a connected semi-reductive algebraic group over an algebraically closed field $\mathbb{F}$ and g=Lie($G$). It turns out that $G$ has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for g, which says that the $G$-invariant ring $\mathbb{F}$[g]^G is a polynomial ring if g satisfies a certain "positivity" condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.

We introduce the $n$-pure projective (resp., injective) dimension of complexes in $n$-pure derived categories, and give some criteria for computing these dimensions in terms of the $n$-pure projective (resp., injective) resolutions (resp., coresolutions) and $n$-pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of $n$-pure global dimension of rings. Finally, we study Verdier quotient of bounded $n$-pure derived category modulo the bounded homotopy category of $n$-pure projective modules, which is called an $n$-pure singularity category since it can reflect the finiteness of $n$-pure global dimension of rings.

Let $U(n,\mathbb{Q})$ be the group of all $n \times n$ (upper) unitriangular matrices over rational numbers field $\mathbb{Q}$. Let $S$ be a subset of $U(n,\mathbb{Q})$. In this paper, we prove that $S$ is a subgroup of $U(n,\mathbb{Q})$ if and only if the $(i,j)$-th entry $S_{ij}$ satisfies some condition (see Theorem 3.5). Furthermore, we compute the upper central series and the lower central series for $S$, and obtain the condition that the upper central series and the lower central series of $S$ coincide.

In this paper, we give an equitable presentation for the multiparameter quantum group associated to a symmetrizable Kac-Moody Lie algebra, which can be regarded as a natural generalization of the Terwilliger's equitable presentation for the one-parameter quantum group.

Let G be a finite group, and let P be a Sylow p-subgroup of G. Under the hypothesis that N_G(P) is p-nilpotent, we provide some conditions to give a p-nilpotency criterion of finite groups by Engel condition, which improves some recent results.

Let G be a classical group over an arbitrary field F, acting on an n-dimensional vector space V = V (n, F) over a field F. In this paper, we classify the maximal subgroups of G, which normalizes a solvable subgroup N of GL(n, F) not lying in F^*1_V.