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.