Nilpotent Matrices Research Articles

Shifted Yangians and finite [formula omitted]-algebras

We give a presentation for the finite W -algebra associated to a nilpotent matrix in the general linear Lie algebra over C . In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l , the presentation is that of the Yangian of level l associated to g l n , as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.

A Lower Bound on the C-Numerical Radius of Nilpotent Matrices Appearing in Coherent Spectroscopy

We provide a lower bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. Mathematically the problem is the determination of the $C$-numerical radius of $A$ for certain nilpotent matrices $C$ and $A$. The presented lower bound is conjectured to be exact as it coincides with numerical data provided in [U. Helmke et al., J. Global Optim., 23 (2002), pp. 283-308].

Nilpotent symmetric Jacobian matrices and the Jacobian Conjecture II

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k n → k n of the form F = x + H , such that JH is nilpotent and symmetric, when n ⩽ 4 . If H is also homogeneous a similar result is proved for all n ⩽ 5 .

Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order- m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation ( m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.

Nilpotent linear transformations and the solvability of power-associative nilalgebras

We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert’s problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.

On nilpotent matrices over distributive lattices

In this paper, nilpotent matrices over distributive lattices are considered. Some properties and characterizations for nilpotent matrices are established and in particular, a necessary and sufficient condition for an n × n nilpotent matrix to have the nilpotent index n is given. Also, some properties of the reduction of a nilpotent matrix are obtained. Partial results obtained in this paper are results on nilpotent fuzzy matrices by H. Hashimoto.

A p-adic local monodromy theorem

We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential equations, analogous to Grothendieck's local monodromy theorem (also a consequence of results of Andre and of Mebkhout). Namely, given a finite locally free sheaf on an open p-adic annulus with a connection and a compatible Frobenius structure, the module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix.

Nilpotent symmetric Jacobian matrices and the Jacobian conjecture

Let H : C n→ C n be a polynomial map such that the Jacobian JH of H is nilpotent and symmetric. The symmetric dependence problem, SDP( n), asks whether the rows of the matrix JH are dependent over C . We show that if SDP( r) has an affirmative answer for all r⩽ n, then the Jacobian conjecture holds for all F : C n→ C n of the form F= x+ H with JH nilpotent and symmetric. As a consequence, we deduce the main result of (J. Pure Appl. Algebra, 189/1–3, 123–133), which asserts that the Jacobian conjecture holds for all polynomial maps of the form F= x+ H, with JH nilpotent, symmetric and homogeneous, and n⩽4.

Nilpotent symmetric Jacobian matrices and the Jacobian conjecture

Let H:Cn→Cn be a polynomial map such that the Jacobian JH of H is nilpotent and symmetric. The symmetric dependence problem, SDP(n), asks whether the rows of the matrix JH are dependent over C. We show that if SDP(r) has an affirmative answer for all r⩽n, then the Jacobian conjecture holds for all F:Cn→Cn of the form F=x+H with JH nilpotent and symmetric. As a consequence, we deduce the main result of (J. Pure Appl. Algebra, 189/1–3, 123–133), which asserts that the Jacobian conjecture holds for all polynomial maps of the form F=x+H, with JH nilpotent, symmetric and homogeneous, and n⩽4.

Irreducible operator semigroups such that AB and BA are proportional

Let S be an irreducible semigroup of compact linear operators on a Banach space of dimension at least 2 with the property that the products AB and BA are proportional for each pair of elements A and B in S . We show that S is necessarily a nilpotent matrix group of nilpotency class 2 with possible addition of the zero matrix. We also study generalizations of the property.

Irreducible operator semigroups such that AB and BA are proportional

Let S be an irreducible semigroup of compact linear operators on a Banach space of dimension at least 2 with the property that the products AB and BA are proportional for each pair of elements A and B in S . We show that S is necessarily a nilpotent matrix group of nilpotency class 2 with possible addition of the zero matrix. We also study generalizations of the property.

On matrix perturbations with minimal leading Jordan structure

We show that any matrix perturbation of an n× n nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. Additionally, we derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskii's perturbation theorem for eigenvalues are necessary too. It is further shown how minimality can be obtained by computing a similarity transform whose entries are polynomials of degree at most n. This relies on an extension of both Lidskii's theorem and its Newton diagram-based interpretation.

On nilpotent fuzzy matrices

In this paper, we study the issue of nilpotent fuzzy matrices. We first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues. When a finite number of fuzzy matrices are simultaneously considered, we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices. On the other hand, it is well known that the max–min algebra is one kind of lattice. We shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice.

On the irreducibility of commuting varieties of nilpotent matrices

Given an n× n nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the n× n nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the pairs ( A, B) of n× n nilpotent matrices over K such that [ A, B]=0 if either char K=0 or char K⩾ n/2. We get as a consequence a proof of the irreducibility of the local Hilbert scheme of n points of a smooth algebraic surface over K if either char K=0 or char K⩾ n/2.

Dimension of the orbit of marked subspaces

Given a nilpotent endomorphism, we consider the manifold of invariant subspaces having a fixed Segre characteristic. In [Linear Algebra Appl., 332–334 (2001) 569], the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained. Here we obtain the explicit form of this deformation when the invariant subspace is marked, and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces (those with open orbit). Moreover, we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace.

On simultaneously nilpotent fuzzy matrices

Nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices. In this paper, we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices. The notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix. Properties of the simultaneous nilpotence will be established. In the study of consecutive powers of a fuzzy matrix, a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix. In this paper, we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices. Similar to the nilpotent characterization for a controllable fuzzy matrix, the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices.

Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.

On nilpotent singular systems

We derive some results for linear differential-algebraic equations (DAEs) of the form N(t) z′(t)+ z(t)= h(t) , where N( t) is a smooth nilpotent matrix for all t concerned and such that the system is uniquely solvable, i.e., has exactly one solution for each smooth h . Such systems play a fundamental role in the investigation of more general DAEs, but their theory is still incomplete. We give some sufficient conditions for unique solvability, and a global representation for the solution operator constructed in terms of a finite set of special solutions.

On the powers of matrices over a distributive lattice

In this paper, the index and the period for a lattice matrix are estimated. Some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained. Also, some equivalent conditions for a nilpotent lattice matrix are discussed.

Elliptic Crystals and Modular Motives

The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves. Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic setting, so that it applies also to semistable (degenerate) elliptic curves. Roughly speaking, an elliptic crystal over a (logarithmic) perfect field k of characteristic p is a free W-module E of rank two equipped with a Frobenius-linear endomorphism 8, a nilpotent linear endomorphism N, a two-step Hodge filtration A on k E, and an isomorphism tr: 4E W. These data are required to satisfy various compatibilities: for example, A(k E) is the kernel of idk 8. When N is not zero, such a crystal is quite rigid, and in particular its canonical coordinates are very precisely determined (1.3). Liftings of elliptic crystals from k to its Witt ring are determined by specifying a lifting B of the Hodge filtration A of k E, and the classification is made explicit in (1.5). Our first main result (2.2) concerns the relationship between deformation theory and elliptic crystals on a curve. Let X k be the reduction modulo p of a smooth log curve Y W over W, let (E, B) be an elliptic crystal on Y W, and let (E, A) be the restriction of (E, B) to X k. Following Mochizuki [8] and Gunning [6], we shall say that E is indigenous if its Kodaira-Spencer mapping is an isomorphism, a condition which can be checked either on Y W or on X k. If f: X$ X is a morphism of smooth log curves over k then a lifting g: Y$ Y of f determines a lifting g*(BEY) ( f *E)Y$ of A( f *E)X$ k . Theorem (2.2) shows that if p is odd and E is indigenous, the resulting map from the set of liftings of f to the set of liftings of the mod p Hodge filtration of f *EX is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of AEX is positive and p is odd) an indigenous bundle on X k determines a unique lifting Y W, characterized by the fact doi:10.1006 aima.2001.1976, available online at http: www.idealibrary.com on