Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

On finite GK-dimensional Nichols algebras over abelian groups

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

A NEW ALGORITHM FOR DECOMPOSING MODULAR TENSOR PRODUCTS

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$ , the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$ . This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$ . Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$ . Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions

In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

Boundary null-controllability of two coupled parabolic equations : simultaneous condensation of eigenvalues and eigenfunctions.

Let the matrix operator $L=D\partial_{xx}+q(x)A_0 $, with $D=diag(1,\nu)$, $\nu\neq 1$, $q\in L^{\infty}(0,\pi)$, and $A_0$ is a Jordan block of order $1$. We analyze the boundary null controllability for the system $y_{t}-Ly=0$. When $\sqrt{\nu} \notin \mathbb{Q}_{+}^*$ and $q$ is constant, $q=1$ for instance, there exists a family of root vectors of $(L^*,\mathcal{D}(L^*))$ forming a Riesz basis of $L^{2}(0,\pi;\mathbb{R}^2 )$. Moreover in \cite{JFA14} the authors show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^*,\mathcal{D}(L^*))$, that is to say the existence of $T_0(\nu)$ such that the system is null controllable at time $T > T_0(\nu)$ and not null controllable at time $T < T_0(\nu)$. In the same paper, the authors prove that for all $\tau \in [0, +\infty]$, there exists $\nu \in ]0, +\infty[$ such that $T_0(\nu)=\tau$. When $q$ depends on $x$, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.

Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial Jordan block case

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit.

Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p

Let V be a vector space over a field of characteristic p. In this paper we complete the classification of all irreducible subgroups G of {\mathrm{GL}(V)} that contain a p-element whose Jordan normal form has exactly one non-trivial block, and possibly multiple trivial blocks. Broadly speaking, such a group acting primitively is a classical group acting on a symmetric power of a natural module, a 7-dimensional orthogonal group acting on the 8-dimensional spin module, a complex reflection group acting on a reflection representation, or one of a small number of other examples, predominantly with a self-centralizing cyclic Sylow p-subgroup.

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.