Lie algebras with few centralizers

We will characterize all finite dimensional Lie algebras with at most |F|²+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cocalibrated G₂-structures and cocalibrated ${{\rm G}_2^*}$ -structures are the natural initial values for Hitchin’s evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin₀(3, 4), respectively. In this article, we classify 7-D real Lie algebras with a codimension one Abelian ideal which admit such structures. Moreover, we classify the 7-D complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated ${({\rm G}_2)_{\mathbb{C}}}$ -structures.     

Abelian subalgebras in some particular types of Lie algebras

It is well-known that there exists a close link between Lie Theory and Relativity Theory. Indeed, the set of all symmetries of the metric in our four-dimensional spacetime is a Lie group. In this paper we try to study this link in depth, by dealing with three particular types of Lie algebras: h_n algebras, g_n algebras and Heisenberg algebras. Our main goal is to compute the maximal abelian dimensions of each of them, which will allow us to move a step forward in the advancement of this subject.     

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to 5 or of dimension greater than or equal to 3 with rank-one.     

Infinite groups with Abelian centralizers of involutions

The article contains two characterizations of projective linear groups PGL₂(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second, in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which are recognizable by the set of their element orders. In memory of Viktor A. Gorbunov Supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 74–86, January–February, 2000.     

Groups with elementary Abelian centralizers of involutions

An involution i of a group G is said to be almost perfect in G if any two involutions of i^G the order of a product of which is infinite are conjugated via a suitable involution in i^G. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 75–82, January–February, 2007.     

On centralizers of reflexive algebras

Let ${\mathcal{L}}$ be a subspace lattice on a complex Banach space X and δ be a linear mapping from ${alg\mathcal{L}}$ into B(X) such that for every ${A \in alg\mathcal{L}, 2\delta(A^2)=\delta(A)A + A\delta(A)}$ or ${\delta(A^3) = A\delta(A)A}$ . We show that if one of the following holds (1) ${\vee\{L : L \in \mathcal{J}(\mathcal{L})\}=X}$ , (2) ${\wedge\{L_-: L \in \mathcal{J}(\mathcal{L})\}=(0)}$ and X is reflexive, then δ is a centralizer. We also show that if ${\mathcal{L}}$ is a CSL and δ is a linear mapping from ${alg\mathcal{L}}$ into itself, then δ is a centralizer.     

Finite algebras with Abelian properties

Presented by R. Quackenbush.     

Extremal generalized centralizers in matrix algebras

We describe matrices with extremal generalized centralizers over algebraically closed fields.     

On Lie algebras associated with representation-directed algebras

Let B be a representation-finite C-algebra. The Z-Lie algebra L(B) associated with B has been defined by Riedtmann in [Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994) 526-546]. If B is representation-directed, there is another Z-Lie algebra associated with B defined by Ringel in [C.M. Ringel, Hall Algebras, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 433-447] and denoted by K(B).We prove that the Lie algebras L(B) and K(B) are isomorphic for any representation-directed C-algebra B.     

Noetherianness of wreath products of Abelian Lie algebras with respect to equations of universal enveloping algebra

It is proved that a wreath product of two Abelian finite-dimensional Lie algebras over a field of characteristic zero is Noetherian w.r.t. equations of a universal enveloping algebra. This implies that an index 2 soluble free Lie algebra of finite rank, too, has this property.     

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.