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Ivan Kozlov

Publications and source records attributed to Ivan Kozlov.

7 recordsLinked to original sources
arXiv2023

Jordan-Kronecker invariants of Lie algebra representations: examples and computations

In these paper we compute Jordan-Kronecker invariants of Lie algebra representations, introduced earlier by A.V. Bolsinov, A.M. Izosimov and I.K. Kozlov, for a number of representations. In particular, we compute them for the sums of standard representations of $\operatorname{gl}(n)$, $\operatorname{sl}(n)$, $\operatorname{so}(n)$, $\operatorname{sp}(n)$, and the Lie algebra of upper triangular matrices $\operatorname{b}(n)$; the standard representation of Lie algebra of strictly upper triangular matrices $\operatorname{n}(n)$; and for the differential of the congruence action of $\operatorname{GL}(n)$ and $\operatorname{SL}(n)$ on symmetric forms and skew-symmetric forms.

math.RT↗
arXiv2023

The topology of Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra $\textrm{so}(4)$, which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank $0$ are determined, the bifurcations of Liouville tori are described and the loop molecules for all singular points of the bifurcation diagram are computed. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra $\textrm{so}(4)$ by taking a natural limit.

math.DG↗
arXiv2019

Jordan-Kronecker invariants of Lie algebra representations and degrees of invariant polynomials

For an arbitrary representation $ρ$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $ρ$. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of $ρ$. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

math.RT↗
arXiv2018

Integral affine 3-manifolds

Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering. Also a complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.

math.DG↗
arXiv2011

Elementary proof of Jordan-Kronecker theorem

In this paper we prove the Jordan-Kronecker theorem which gives a canonical form for a pair of skew-symmetric bilinear forms on a finite-dimensional vector space over an algebraically closed field.

math.RA↗