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Mikhail Ignatyev

Publications and source records attributed to Mikhail Ignatyev.

6 recordsLinked to original sources
arXiv2021

The geometric classification of $2$-step nilpotent algebras and applications

We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.

math.RA↗
arXiv2020

Reconstruction formula for differential systems with a singularity

Our studies concern some aspects of scattering theory of the singular differential systems $ y'-x^{-1}Ay-q(x)y=ρBy, \ x>0 $ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $ρ$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl - type solutions of the considered differential system. Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation, provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula we establish first the asymptotical expansions for the Weyl - type solutions as $ρ\to\infty$ with $o\left(ρ^{-1}\right)$ rate remainder estimate.

math.SP↗
arXiv2016

Spectral analysis for differential systems with a singularity

We consider the differential system $y'-x^{-1}Ay-q(x)y=ρBy $ with $n\times n$ matrices $A,B, q(x)$, where $A,B$ are constant, $B$ is diagonal, $A$ and $q(x)$ are off-diagonal, $q(\cdot)\in W^1_1[0,\infty)$. Some distinguished fundamental system of solutions is constructed. Also, we discuss the inverse scattering problem and obtain the uniqueness result.

math.SP↗
arXiv2015

Infinite Kostant cascades and centrally generated primitive ideals of $U(\mathfrak{n})$ in types $A_{\infty}$, $C_{\infty}$

We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$, $\mathfrak{so}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$. There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we provide explicit generators of the center of $U(\mathfrak{n})$ in all cases. We then fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$ and characterize the centrally generated primitive ideals of $U(\mathfrak{n})$ for $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$ in terms of the above generators. As a preliminary result, we provide a characterization of the centrally generated primitive ideals in the enveloping algebra of the nilradical of a Borel subalgebra of $\mathfrak{sl}_n(\mathbb{C})$, $\mathfrak{sp}_{2n}(\mathbb{C})$.

math.RT↗
arXiv2013

Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result

In a finite-dimensional Euclidian space we consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering problem for Sturm-Liouville differential operator with standard matching conditions in the internal vertices. Transport, spectral and scattering problems for differential operators on graphs appear frequently in mathematics, natural sciences and engineering. In particular, direct and inverse problems for such operators are used to construct and study models in mechanics, nano-electronics, quantum computing and waveguides. The most complete results on (both direct and inverse) spectral problems were achieved in the case of Sturm-Liouville operators on compact graphs, in the noncompact case there are no similar general results. In this paper, we establish some properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the scattering data. A uniqueness theorem for such inverse problem is proved.

math.SP↗