Abstract. In this paper we classify a linear family of Lie brackets on the space of rectangular matrices Mat(n×m, K) and we give an analogue of the Ado’s Theorem. We give also a similar classification on the algebra of the square matrices Mat(n, K) and as a consequence, we prove that we can’t built a faithful representation of the (2n + 1)-dimensional Heisenberg Lie algebra Hn in a vector space...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is ‘almost maximal’ (that is, ...

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2 : n;ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2 : n;ω2) with a Block algebra. W...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that ...

The second homology group H2(st(m,n,R)) of the Steinberg Lie superalgebra st(m,n,R)(m + n ≥ 5), which is trivial, has been studied by A.V.Mikhalev and I.A.Pinchuk in [MP]. In this paper, we will work out H2(st(m,n,R)) explicitly for m + n = 3, 4 which is also trivial when m + n = 3, but not necessarily trivial when m + n = 4. Introduction Steinberg Lie algebras stn(R) and/or their universal cov...

In this paper, we study the Cartan invariants for rank-one Cartan type Lie algebra W (1, n). This is done by reducing representations of generalized restricted Cartan type Lie algebra W (1, n) to representations of restricted Lie algebras W (1, 1) and of sl(2), and then extending Feldvoss-Nakano’s argument on W (1, 1) to the case W (1, n). Our discussion also shows some surprising difference be...

let $mathcal{a}$ be a $c^*$-algebra and $z(mathcal{a})$ the‎ ‎center of $mathcal{a}$‎. ‎a sequence ${l_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{a}$ with $l_{0}=i$‎, ‎where $i$ is the‎ ‎identity mapping‎ ‎on $mathcal{a}$‎, ‎is called a lie higher derivation if‎ ‎$l_{n}[x,y]=sum_{i+j=n} [l_{i}x,l_{j}y]$ for all $x,y in  ‎mathcal{a}$ and all $ngeqslant0$‎. ‎we show that‎ ‎${l_{n}}_{n...

Let Pn = K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero and div0n (respectively, div c n) be the Lie algebra of derivations of Pn with zero (respectively, constant) divergence. We prove that AutLie(div 0 n) ≃ AutK−alg(Pn) (n ≥ 2) and AutLie(div c n) ≃ AutK−alg(Pn). The Lie algebra div c n is a maximal Lie subalgebra of DerK(Pn). Minimal finite sets of generators...

We consider simply connected, 2-step nilpotent Lie groups N, all of which are diffeomorphic to Euclidean spaces via the Lie group exponential map exp : ˆ → N. We show that every such N with a suitable left invariant metric is the base space of a Riemannian submersion ρ : N* → N, where the fibers of ρ are flat, totally geodesic Euclidean spaces. The left invariant metric and Lie algebra of N* ar...

The aim of this paper is to explain the bialgebra structure of the positive part of the quantized universal enveloping algebra (Drinfeld-Jimbo quantum group) of type Anusing the Lie algebra theory concepts. Recently has been introduced a generalization of Lie algebras, the basic T -Lie algebras [1]. Using the T -Lie algebra concept some new (we think) quantum groups of type Ancan be constructed...