Let R be an Archimedean partially ordered ring in which the square of every element is positive, and N(R) the set of all nilpotent elements of R. It is shown that N(R) is the unique nil radical of R, and that N(R) is locally nilpotent and even nilpotent with exponent at most 3 when R is 2-torsion-free. R is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator...

We study the normalization of integrable analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. In particular, we show that a formally linearizable analytic vector field with a nilpotent linear part is linearizable by convergent transformatio...

Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier–Sato transform on the Lie algebra followed by re...

In this paper, we prove results concerning the large scale geometry of connected, simply connected nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT0 metric space or an Alexandrov metric space. The main technical aspect of this work is the proof of a limited...

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying π0 is the theory of ordinary n-nilpotent groups and that t...

Let c ≥ 0, d ≥ 2 be integers and N (d) c be the variety of groups in which every dgenerator subgroup is nilpotent of class at most c. N.D. Gupta posed this question that for what values of c and d it is true that N (d) c is locally nilpotent? We prove that if c ≤ 2 d + 2 − 3 then the variety N (d) c is locally nilpotent and we reduce the question of Gupta about the periodic groups in N (d) c to...

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces trM M and trM M are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5 5 nilpotent matrix has circular numerical ran...

Abstract. Let Bn be the group of upper-triangular invertible n×n matrices and Xn be the variety of strictly upper triangular n × n matrices of nilpotent order 2. Bn acts on Xn by conjugation. In this paper we describe geometry of orbits Xn/Bn in terms of link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotent order 2 and intersectio...