arXiv subjects

Dmitry V. Millionschikov

Publications and source records attributed to Dmitry V. Millionschikov.

4 recordsLinked to original sources

arXiv2017

Naturally graded Lie algebras (Carnot algebras) of slow growth

A nilpotent Lie algebra ${\mathfrak g}$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra ${\rm gr} \mathfrak{g}$ with respect to filtration by ideals of the lower central series. This concept is equivalent to the concept of the Carnot algebra arising in sub-Riemannian geometry and the geometric control theory. We classify finite-dimensional and infinite-dimensional naturally graded Lie algebras (Carnot algebras) ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ For growth functions of such Lie algebras, we have the estimate $F(n) \le \frac{3}{2}n{+}1$.

math.RA↗

arXiv2017

Lie algebras of slow growth and Klein-Gordon equation

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra $χ$. We establish two explicit isomorphisms between characteristic Lie algebras of sinh-Gordon and Tzitzeica equations and pro-solvable Lie subalgebras of affine Kac-Moody algebras $A_1^{(1)}$ and $A_2^{(2)}$ respectively. Hence both characteristic Lie algebras are slowly linearly growing Lie algebras with average growth rates $\frac{3}{2}$ and $\frac{4}{3}$ respectively.

math.RA↗

arXiv2017

Graded Thread Modules over the Positive Part of the Witt (Virasoro) Algebra

We study ${\mathbb Z}$-graded thread $W^+$-modules $$V=\oplus_i V_i, \; \dim{V_i}=1, -\infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\infty, N=\infty$) two-parametric family $V_{λ, μ}$ of $W^+$-modules induced by the twisted $W$-action on tensor densities $P(x)x^μ(dx)^{-λ}, μ, λ\in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{α, β}$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=αf_{i+1}$ and $e_2f_j=βf_{j+2}$ for $i,j \in {\mathbb Z}$ and $α, β$ are two arbitrary constants ($e_if_j=0, i \ge 3$). We classify $(n+1)$-dimensional graded thread $W^+$-modules for $n$ sufficiently large $n$ of three important types. New examples of graded thread $W^+$-modules different from finite-dimensional quotients of $V_{λ, μ}$ and $C_{α, β}$ were found.

math.RT↗

arXiv2009

The Variety of Lie algebras of maximal class

We present an explicit description of the affine variety of Lie algebras of the maximal class (filiform Lie algebras): the formulas of polynomial equations that determine this variety are written. It can considered as the base of the nilpotent versal deformation of N-graded Lie algebra m_0.

math.RA↗