For every finite $p$-group $G$ of order $p^n$ with derived subgroup of order $p^m$, Rocco in \cite{roc} proved that the order of tensor square of $G$ is at most $p^{n(n-m)}$. This upper bound has been improved recently by author in \cite{ni}. The aim of the present paper is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given $n$-dimensional non-abelian nilpotent Lie algebra $L$ with derived subalgebra of dimension $m$ we have $\mathrm{dim} (L\otimes L)\leq (n-m)(n-1)+2$. Furthermore for $m=1$, the explicit structure of $L$ is given when the equality holds.
Let $L/K$ be a finite Galois extension of fields with group $Γ$. When $Γ$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $Γ$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $Γ$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $Γ$. We also give a criterion on $n$ such that \emph{every} Hopf-Galois structure on a cyclic extension of degree $n…
For $c\geq 2$, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set $X$ is the quotient $L/[(L')^c,L]$ where $L$ is the free Lie ring on $X$, and $(L')^c$ denotes the $c$th term of the lower central series of the derived ideal $L'=L^2$ of $L$. In this paper we give a complete description of the torsion subgroup of its additive group in the case where $|X|=2$ and $c$ is a prime number.
We address the question of the dualizability of nilpotent Mal'cev algebras, showing that nilpotent finite Mal'cev algebras with a non-abelian supernilpotent congruence are inherently non-dualizable. In particular, finite nilpotent non-abelian Mal'cev algebras of finite type are non-dualizable if they are direct products of algebras of prime power order. We show that these results cannot be generalized to nilpotent algebras by giving an example of a group expansion of infinite type that is nilpotent and non-abelian, but dualizable. To our knowledge this is the first construction of a non-abelia…
We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U(g)-modules are locally Artinian are precisely those whose even part g_0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.
Let $ L $ be an $ n $-dimensional non-abelian nilpotent Lie algebra and $ s(L)=\frac{1}{2}(n-1)(n-2)+1-\dim \mathcal{M}(L) $ where $ \mathcal{M}(L) $ is the Schur multiplier of a Lie algebra $ L. $ The structures of nilpotent Lie algebras $ L $ when $ s(L)\in \lbrace 0,1,2,3,4,5\rbrace $ are determined. In this paper, we classify all non-abelian nilpotent Lie algebras $ L $ when $ s(L)=6,7. $
Let $L$ be a nilpotent Lie superalgebra of dimension $(m\mid n)$ and $s(L) = \frac{1}{2}[(m + n - 1)(m + n -2)]+ n+ 1 - \dim \mathcal{M}(L)$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Here $s(L)\geq 0$ and the structure of all non-abelian nilpotent Lie superalgebras with $s(L)=0$ is known((\cite{Nayak2019}). This paper is devoted to obtain all nilpotent Lie superalgebras when $s(L) \leq 2$. Further, we apply those results to list all non-abelian nilpotent Lie superalgebras $L$ with $ t(L) \leq 4$.
In this article, we define the capable pairs of Lie superalgebras. We classify all capable pairs of abelian and Heisenberg Lie superalgebras. After that we discuss on pairs of Lie superalgebras with derived subalgebra of dimension one and a non-abelian ideal. Finally, we determine the structure of the Schur multiplier of pairs of Heisenberg Lie superalgebras.
We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n less than 8. Furthermore we study the case where there exists an abelian subalgebra of codimension 2. Here we explicitly construct an abelian ideal of codimension 2 in case of nilpotent Lie algebras.
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n = 1 or the center of D contains at least five elements. This paper contains two main results. In the first one we prove that each nilpotent maximal subgroup of N is abelian; this generalizes the result in [R. Ebrahimian, J. Algebra 280 (2004) 244 - 248] (which asserts that each max- imal subgroup of GLn(D) is abelian) and a result in [M. Ramezan-Nassab, D. Kiani, J. Algebra 376 (2013) 1 - 9]. In the second one we show that a maximal subgroup of GLn(D) cannot be polycyclic-by-finite.
Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …
In this article we define the $c$-nilpotent multiplier of a finite dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent multiplier of finite dimensional nilpotent Lie superalgebras whose derived subalgebras have dimension at most one. Then we give an upper bound on the dimension of $2$-nilpotent multiplier of any finite dimensional nilpotent Lie superalgebra. Moreover, we discuses the $2$-capability of special as well as odd Heisenberg Lie superalgebras and abelian Lie superalgebras.
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.
We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of $n$-dimensional characteristically nilpotent filiform Lie algebras $\mathfrak{f}_n$, for all $n\ge 13$, all of whose derivations are almost inner. Finally we compare the almost inner derivations of Lie algebras considered over two different fields $K\supseteq…
Let Z<X> be the free unitary associative ring freely generated by an infinite countable set X = {x_1, x_2,...}. Define a left-normed commutator [x_1, x_2, ..., x_n] by [a,b] = ab - ba, [a,b,c] = [[a,b],c]. For n \ge 2, let T^(n) be the ideal in Z<X> generated by all commutators [a_1,a_2,..., a_n] (a_i \in Z<X>). It can be easily seen that the additive group of the quotient ring Z<X> /T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z<X> /T^(3) is free abelian as well. In the present note we show that this is not the cas…
Using cocycle twists for associative graded algebras, we characterize finite dimensional nilpotent Lie color algebras $L$ graded by arbitrary abelian groups whose enveloping algebras $U(L)$ have the property that the injective hulls of simple right $U(L)$-modules are locally Artinian. We also collect together results on gradings on Lie algebras arising from this characterization.
We study the algebraic constraints on the structure of nilpotent Lie algebra $\mathbb{g}$, which arise because of the presence of an integrable complex structure $J$. Particular attention is paid to non-abelian complex structures. Constructed various examples of positive graded Lie algebras with complex structures, in particular, we construct an infinite family $\mathfrak{D}(n)$ of such algebras that we have for their nil-index $s(\mathfrak{D}(n))$: $$ s(\mathfrak{D}(n))=[ \frac{2}{3}\dim{\mathfrak{D}(n)} ]. $$
We report on some computations with nilpotent orbits in simple Lie algebras of exceptional type within the SLA package of GAP4. Concerning reachable nilpotent orbits our computations firstly confirm the classification of such orbits in Lie algebras of exceptional type by Elashvili and Grelaud, secondly they answer a question by Panyushev, and thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types. The second topic of this note concerns abelianizations of centralizers of nilpotent elements. We give tables with their dime…
Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nil…
The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the p…
For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.
Let $\mathbb Z \langle X \rangle$ be the free unital associative ring freely generated by an infinite countable set $X = \{ x_1,x_2, \dots \}$. Define a left-normed commutator $[x_1,x_2, \dots, x_n]$ by $[a,b] = ab - ba$, $[a,b,c] = [[a,b],c]$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $\mathbb Z \langle X \rangle$ generated by all commutators $[a_1,a_2, \dots, a_n]$ $( a_i \in \mathbb Z \langle X \rangle )$. Let $T^{(3,2)}$ be the two-sided ideal of the ring $\mathbb Z \langle X \rangle$ generated by all elements $[a_1, a_2, a_3, a_4]$ and $[a_1, a_2] [a_3, a_4, a_5]$ $(a_i \in \…
To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of cl…
We classify the nilpotent Lie algebras of real dimension eight and minimal center that admit a complex structure. Furthermore, for every such nilpotent Lie algebra $\mathfrak{g}$, we describe the space of complex structures on $\mathfrak{g}$ up to isomorphism. As an application, the nilpotent Lie algebras having a non-trivial abelian $J$-invariant ideal are classified up to eight dimensions.
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 discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.
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.
We look at the odd nilpotent orbits of osp(2n+1,2n), giving a combinatorial interpretation which enables us, via the square map, to explain the link with even nilpotent orbits. We then study the closure ordering of the odd nilpotent orbits. Finally, we give a desingularization of the odd nilpotent cone.
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.
