arXiv subjects
Publications and source records attributed to Peyman Niroomand.
Let $L$ be an $(m\vert n)$-dimensional nilpotent Lie superalgebra where $m + n \geq 4$ and $n \geq 1$. This paper classifies such nilpotent Lie superalgebras $L$ with a derived subsuperalgebra of dimension $m+n-2$ such that $γ(L) = m + 2n - 2 - \dim \mathcal{M}(L)$, where $γ(L) \in \{0, 1, 2\}$ and $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Furthermore, we show that all these superalgebras are capable.
Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $ p^{\frac{1}{2}(d-1)(n-k+2)+1}. $ In the current paper, we find the structure of all $p$-groups that attains the mentioned bound. Moreover, we show that all of them are capable.
It is known that the dimension of the Schur multiplier of a non-abelian nilpotent Lie algebra $L$ of dimension $n$ is equal to $\frac{1}{2}(n-1)(n-2)+1-s(L)$ for some $ s(L)\geq0 $. The structure of all nilpotent Lie algebras has been given for $ s(L) \leq 4 $ in several papers. Here, we are going to give the structure of all non-abelian nilpotent Lie algebras for $s(L)=5$.
In this paper, we give the explicit structure of $ \otimes^{3} H $ and $ \wedge^{3} H $ where $ H $ is a generalized Heisenberg Lie algebra of rank at most $ 2. $ Moreover, for a non-abelian nilpotent Lie algebra $ L, $ we obtain an upper bound for the dimension of $ \otimes^{3} L.
Recently, the authors obtained the Schur multiplier, the non-abelian tensor square and the non-abelian exterior square of $d$-generator generalized Heisenberg Lie algebras of rank $ \frac{1}{2}d(d-1).$ Here, we intend to obtain the same results for $d$-generator generalized Heisenberg Lie algebras of rank $ t$ when $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$ Then, as a result, we give similar consequences for a nilpotent Lie algebra $ L $ of class two when $ \dim (L/Z(L))=d,$ $ \dim L^2=t $ such that $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$
The paper is devoted to give a full classification of all finite dimensional nilpotent Lie algebras $ L $ of class $4$ such that $ \dim L^2=3. $ Moreover, we classify the capable ones.
Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\frac{1}{2}d(d-1)-1}$ for $ d\geq 3 $ and $ p\neq 2. $ Additionally, when $ G $ is of exponent $ p, $ we give the structure of $ G. $
There are some results on nilpotent Lie algebras $ L $ investigate the structure of $ L $ rely on the study of its $2$-nilpotent multiplier. It is showed that the dimension of the $2$-nilpotent multiplier of $ L $ is equal to $ \frac{1}{3} n(n-2)(n-1)+3-s_2(L).$ Characterizing the structure of all nilpotent Lie algebras has been obtained for the case $ s_2(L)=0.$ This paper is devoted to the characterization of all nilpotent Lie algebras when $ 0\leq s_2(L)\leq 6.$ Moreover, we show that which of them are $2$-capable.
The power graph of an arbitrary group $G$ is a simple graph with all elements of $G$ as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices are all elements of $G$ that generate a proper subgroup of $G$ and two elements are adjacent if the cyclic subgroup generated by which have non-trivial intersections. We concentrate on completeness and planarity of this graph.
In this paper, we extend the notion of the Bogomolov multipliers and the CP-extensions to Lie algebras. Then we compute the Bogomolov multipliers for Abelian, Heisenberg and nilpotent Lie algebras of class at most 6. Finally we compute the Bogomolov multipliers of some simple complex Lie algebras.
In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.
By considering the nilpotent Lie algebra with the derived subalgebra of dimension $ 2$, we compute some functors include the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.
In this paper, we determine the behavior of the $c$-nilpotent multiplier of Lie algebras with respect to the direct sums. Then we give some results on the $c$-capability of the direct sum of finite dimensional Lie algebras.
The paper is devoted to obtain an upper bound for the Schur multiplier of nilpotent Lie algebras of maximal class. It improves the later upper bounds on the Schur multiplier of such Lie algebras.
The aim of this work is to find some exact sequences on the $c$- nilpotent multiplier of a group $G$. We also give an upper bound for the $c$- nilpotent multiplier of finite $p$-groups and give the explicit structure of groups whose take the upper bound. Finally, we will get the exact structure of the $c$- nilpotent multiplier and determine $c$-capable groups in the class of extra-special and generalized extra-special $p$-groups. It lets us to have a vast improvement over the last results on this topic.
From [Problem 1729, Groups of prime power order, Vol. 3], Berkovich et al. asked to obtain the Schur multiplier and the representation of a group $G$, when $G$ is a special $p$-group minimally generated by $d$ elements and $|G'|=p^{\frac{1}{2}d(d-1)}$. Since there are analogies between groups and Lie algebras, we intend to give an answer to this question similarly for nilpotent Lie algebras. Furthermore, we give some results about the tensor square and the Schur multiplier of some nilpotent Lie algebras of class two.
For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible.
For every $p$-group of order $p^n$ with the derived subgroup of order $p^m$, Rocco in \cite{roc} has shown that the order of tensor square of $G$ is at most $p^{n(n-m)}$. In the present paper not only we improve his bound for non-abelian $p$-groups but also we describe the structure of all non-abelian $p$-groups when the bound is attained for a special case. Moreover, our results give as well an upper bound for the order of $π_3(SK(G, 1))$.