In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pai...

The Jordan canonical form parametrises similarity classes in the nilpotent cone Nn, consisting of n× n nilpotent complex matrices, by partitions of n. Achar and Henderson (2008) extended this and other well-known results about Nn to the case of the enhanced nilpotent cone C ×Nn. 1. Jordan canonical form The Jordan canonical form (JCF), introduced in 1870 [10], is one of the most useful tools in...

Lemma 8.4 If C is a n n × matrix with 0 det ≠ C , then, there exists a n n × (complex) matrix B such that C e = . Proof: For any matrix C , there exists an invertible matrix P , s.t. 1 P CP J − = , where J is a Jordan matrix. If C e = , then, 1 1 1 P B P B e P e P P CP J − − − = = = . Therefore, it is suffice to prove the result when C is in a canonical form. Suppose that 1 ( , , ) s C diag C C...

This paper contains the factorization of the polyphase matrix of finite impulse response perfect reconstruction filter banks into unimodular factors containing finite Jordan nilpotent structures and the associated transform matrices. An important contribution of the paper is the proposal of a systematic procedure for the construction of the transform matrices. The factorization is based on the ...

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

Tits has shown that a finitely generated matrix group either contains a nonabelian free group or has a solvable subgroup of finite index. We give a polynomial time algorithm for deciding which of these two conditions holds for a given finitely generated matrix group over an algebraic number field. Noting that many computational problems are undecidable for groups with nonabelian free subgroups,...

for Ak−1 models with open boundaries, by constructing polynomial solutions of level one boundary quantum Knizhnik–Zamolodchikov equations for Uq(sl(k)). The result takes the form of a character of the symplectic group, that leads to a generalization of the number of vertically symmetric alternating sign matrices. We also investigate the other combinatorial point q = −1, presumably related to th...

This paper develops a method for constructing nilpo-tent approximations for local representations of invariant systems on matrix Lie groups via a simple operation on the structure constants of the associated Lie algebra. The crucial role such nilpotent approximations play for the problem of feedback nilpotentization is discussed. The presented ideas are illustrated with an example modeling the ...

τ(w)(x) = 〈x(w), w〉, w ∈ W, x ∈ g ⊆ End(W ), and similarly for τ ′. Our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps. It is easy to see that for a nilpotent coadjoint orbit O ⊆ g∗ the set τ ′(τ−1(O)) is the union of nilpotent coadjoint orbits in g′. It turns out that it is a closure of a single orbit: Theorem 1.1 Let O ⊆ g∗ be a nilpotent ...

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector. Introduction Throughout this paper V is a vector space of positive dimension over a field f and g is a nilpotent Lie algebra over f of linear operators on V . An element u ∈ V is an eigenvector for S ⊂ g if u is an eigenvector for every o...