arXiv subjects
Publications and source records attributed to Dmitri V. Millionschikov.
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).
We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $ω$ (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 $d_ω=d +λω$. Using Witten's approach to the Morse theory one can estimate the number of critical points of $ω$ in terms of the cohomology of deformed de Rham complex with sufficiently large values of $λ$ (torsion-free Novikov's inequalities). We show that for an interesting class of solvmanifolds this cohomology can be computed as the cohomology of the corresponding Lie algebra $\mathfrak{g}$ associated with the one-dimensional representation $ρ_{λω}$.
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.
We study symplectic structures on filiform Lie algebras -- nilpotent Lie algebras of the maximal length of the descending central sequence. In the present article we classify the Lie algebras with the structure relations of the following form: $$[e_i,e_j]=(j-i)e_{i+j}+\sum_{l{=}1}c_{ij}^l e_{i+j+l}, i+j \le n.$$ In even dimensions the subspace of symplectic Lie algebras has the codimension one.
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 describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplectic filiform Lie algebra is a filtered deformation of some N-graded symplectic filiform Lie algebra. But this condition is not sufficient. A spectral sequence is constructed in order to answer the question whether a given deformation of a N-graded symplectic filiform Lie algebra admits a symplectic structure or not. Other applications and examples are discussed.
We study the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ of the deRham complex $Λ^*(G/Γ)\otimes{\mathbb C}$ of a compact solvmanifold $G/Γ$ with a deformed differential $d_{λω}=d + λω$, where $ω$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with a completely solvable Lie algebra $\mathfrak{g}$ and a cocompact lattice $Γ\subset G$ the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ coincides with the cohomology $H^*_{λω}(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ associated with the one-dimensional representation $ρ_{λω}: \mathfrak{g} \to {\mathbb K}, ρ_{λω}(ξ) = λω(ξ)$. Moreover $H^*_{λω}(G/Γ, {\mathbb C})$ is non-trivial if and only if $-λ[ω]$ belongs to the finite subset $\{0\} \cup \tilde Ω_{\mathfrak{g}}$ in $H^1(G/Γ, {\mathbb C})$ well defined in terms of $\mathfrak{g}$.