Dmitry V. Millionshchikov

Lomonosov Moscow State University | MSU · Faculty of Mechanics and Mathematics

This research paper presents a new fundamental approach for evaluating accurate ab initio quartic, sextic, and octic centrifugal distortion parameters of A-reduced rotational effective Hamiltonians of asymmetric top molecules. In this framework, the original Watson Hamiltonian, expanded up to sextic terms of kinetic and potential energies, is subje...

Для произвольной вещественной нильпотентной алгебры Ли (нильмногообразия) c интегрируемой комплексной структурой предложен алгоритм построения ее специальной модели с учетом комплексной структуры. В качестве основного приложения получена классификация восьмимерных $2$-порожденных нильпотентных алгебр Ли, которые допускают интегрируемую комплексную...

Two one-parameter families of positively graded Lie superalgebras generated by two elements and two relations that are narrow in the sense of Zelmanov and Shalev are considered. The first family contains the positive part R+ of the Ramon algebra, the second one contains the positive part NS+ of the Neveu-Schwarz algebra. The results of the article...

Models based on sets of effective vibrational–rotational Hamiltonians, parameterized by a small number of spectroscopic reduced quartic, sextic, and octic constants, can conveniently describe highly complex vibrational–rotational spectra of free small-atomic molecules containing tens and hundreds of thousands of observable lines in the microwave an...

The high-order Rayleigh-Schrödinger perturbation theory (RSPT) can be applied for studying anharmonic vibrational problem formulated with the isomorphic Hougen Hamiltonian, but the resulting series usually possess slowly convergent or even divergent behaviour. This flaw can be overcome by the resummation of such series with the multi-valued Hermite...

The operator canonical perturbation theory (CPT) is an efficient tool for solving the molecular vibration-rotation Schrödinger equation. The corresponding Watson Hamiltonian can be written using angular momentum cylindrical ladder operators (Jz,J±=Jx∓iJy) possessing the Lie algebra su(2) commutation relations [J+,J−]=2Jz, [Jz,J±]=±J±. The reduced e...

Показано, что алгебра Ли фонарщика $\mathfrak l$ над полем рациональных чисел, введенная в работах С.О. Иванова, Р.В. Михайлова и А.А. Зайковского, изоморфна бесконечномерной естественно градуированной алгебре Ли максимального класса $\mathfrak m_0$. И. Феликс и А. Мурильо доказали бесконечномерность ее $q$-мерных гомологий $H_q(\mathfrak l,\mathbb...

In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley–Reisner r...

The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrödinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associ...

In this survey, we discuss two research areas related to Massey's higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley-Reisner r...

A pro-nilpotent Lie algebra is said to be naturally graded if it is isomorphic to its associated graded Lie algebra with respect to the filtration by the ideals in the lower central series. Finite-dimensional naturally graded Lie algebras are known in sub-Riemannian geometry and geometric control theory, where they are called Carnot algebras. We cl...

Предложен рекуррентный и монотонный способ построения и классификации нильпотентных алгебр Ли путем последовательных центральных расширений. Он заключается в вычислении вторых когомологий $H^2(\mathfrak g,\mathbb K)$ расширяемой нильпотентной алгебры Ли $\mathfrak g$ с последующим изучением геометрии пространства орбит действия группы автоморфизмов...

Про-нильпотентная алгебра Ли $\mathfrak g$ называется естественно градуированной, если она изоморфна своей ассоциированной градуированной алгебре Ли $\operatorname{gr} \mathfrak{g}$ относительно фильтрации идеалами нижнего центрального ряда. Конечномерные естественно градуированные алгебры Ли известны в субримановой геометрии и геометрической теори...

We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study of the orbit space geometry of the automorphism group action on Grassmannians defined in terms...

We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. The method consists in calculating the second cohomology \(H^{2}(\mathfrak{g}, \mathbb{K})\) of an extendable nilpotent Lie algebra \(\mathfrak{g}\) followed by studying the geometry of the orbit space of the...

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: 1) the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u...

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial gr...

We study \({\mathbb Z}\)-graded thread \(W^+\)-modules \( V=\oplus _i V_i, \, \dim {V_i}=1, -\infty \le k< i < N\le +\infty , \, \dim {V_i}=0, \, {\mathrm{otherwise}}, \) 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_{\lambda...

We classify real and complex infinite-dimensional narrow positively graded Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ In the proof of the main theorem we apply successive c...

We classify real and complex infinite-dimensional narrow positively graded Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ In the proof of the main theorem we apply successive c...

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 $\chi$. We establish two explicit isomorphisms between characteristic Lie algebras of sinh-Gordon and Tzitzeica equations and pro-solvable Lie subalgebras...

Suppose that an infinite-dimensional Lie algebra ${\mathfrak g}$ over the ground field ${\mathbb K}$ is generated multiplicatively by its finite-dimensional subspace $V_1$. For $n>1$ we denote by $V_n$ the ${\mathbb K}$-linear span of all products in elements of $V_1$ of length at most $n$ with arbitrary arrangements of brackets. We define the grow...

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 $V_{\lambda, \mu}$ of $W^+$-...

Мы представляем явную формулу для серии $S_{2,p}(t)$ особых векторов модулей Верма над алгеброй Вирасоро.

We present an explicit formula for the series S2,p(t) of Virasoro singular vectors.

We study the algebraic constraints on the structure of nilpotent Lie algebra $\mathbb{g}$, which arise because of the presence of an integrable complex structure $J$. Particular attention is paid to non-abelian complex structures. Constructed various examples of positive graded Lie algebras with complex structures, in particular, we construct an in...

We present an explicit formula for a new family of Virasoro singular vectors. As a corrolary we get formulas for differentials of Feigin-Fuchs-Rocha-Carridi-Wallach resolution of the the positive nilpotent part of Virasoro (or Witt) algebra $L_1$.

We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is so-called ”positive part” L1 of the Witt algebra W. Buchstaber conjectured that H ∗ (L1) is generated with respect to non-trivial Massey products by H¹(L1). Feigin, Fuchs and Retakh represented H ∗ (L1) by trivial Massey products and the second part of the...

We consider the Lie algebra L(1) of formal vector fields on the line which vanish at the origin together with their first derivatives. V.M. Buchstaber and A.V. Shokurov showed that the universal enveloping algebra U(L(1)) is isomorphic to the Landweber-Novikov algebra S tensored with the reals. The cohomology H*(L(1)) = H*(U(L(1))) was originally c...

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.

We present an explicit description of the affine variety M Fil of Lie algebras of maximal class (filiform Lie algebras): the formulas of polynomial equations that determine this variety are written down. The affine variety M Fil can be considered as the base of the nilpotent versal deformation of the ℕ-graded Lie algebra m0.

We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform Lie algebras) m0 and m2. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from this list is the “positive” part L 1 of the Witt (or Virasoro) algebra, and its a...

We consider the Lie algebra L 1 of formal vector fields on the line which vanish at the origin together with their first derivatives. V. M. Buchstaber and A. V. Shokurov showed that the universal enveloping algebra U(L 1) is isomorphic to the Landweber-Novikov algebra S tensored with the reals. The cohomology H*(L 1) = H*(U(L 1)) was originally cal...

We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and...

We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is the positive part L_1 of the Witt algebra $W$. We consider an associated graded algebra m_0 of L_1 with respect to the descending central series and prove that H*(m_0) is generated with respect to non-trivial Massey products by one cohomology H^1(m_0).

It was shown by A. Fialowski that an arbitrary infinite-dimensional N-graded “filiform type” Lie algebra with one-dimensional homogeneous components gi such that [g1,gi]=gi+1,∀i⩾2 over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: m0,m2,L1, where the Lie algebras m0 and m2 are defined by their...

We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $\omega$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold $M^n$ with a deformed differential $d_{\omega}=d +\lambda \omega$. Using Witten's appr...

We study symplectic structures on filiform Lie algebras, which are niplotent Lie algebras with the maximal length of the descending central sequence. Let g be a symplectic filiform Lie algebra and dim g = 2k ≥ 12. Then g is isomorphic to some ℕ-filtered deformation either of m0(2k) (defined by the structure relations [e 1, e i ] = e i+1, i = 2,…,...

The cohomology H*(lambdaomega) (G/Gamma, C) of the de Rham complex Lambda* (G/Gamma) circle times C of a compact solvmanifold G/Gamma with deformed differential d(lambdaomega) = d + lambdaomega, where omega is a closed 1-form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any, completely solvable Lie g...

We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.

Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints

We study symplectic structures on filiform Lie algebras – nilpotent Lie algebras of the maximal length of the descending central sequence. There are two basic examples of symplectic Z>0-graded filiform Lie algebras defined by their basises e1,..., e2k and structure relations 1) m0(2k) : [e1, ei] = ei+1, i = 2,..., 2k−1. 2) V2k: [ei, ej] = (j−i)ei+j...

We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie algebras). In particular we de...

Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints

We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$, where $\omega$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that fo...

Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints