A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\mathrm{SCAP}$-subalgebra if there is a chief series $0=L_0\subset L_1\subset ...\subset L_t=L$ of $L$ such that for every $i=1,2,...,t$, we have $H+L_i=H+L_{i-1}$ or $H\cap L_i=H\cap L_{i-1}$. This is analogous to the concept of $\mathrm{SCAP}$-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $\mathrm{SCAP}$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

We introduce a 1-cocycle on the group of diffeomorphisms Diff(M) of a smooth manifold M endowed with a projective connection. This cocycle represents a nontrivial cohomology class of Diff(M) related to the Diff(M)-modules of second order linear differential operators on M. In the one-dimensional case, this cocycle coincides with the Schwarzian derivative, while, in the multi-dimensional case, it represents its natural and new generalization. This work is a continuation of [3] where the same problems...

We give a general definition of branched, self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk’s and Gupta–Sidki’s torsion groups are nil as well as self-similar.We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.

We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a non-degenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, which is analogous to the classical structure of the de Rham cohomology in Kähler...

We study a family of commuting selfadjoint operators $={\left(A_k\right)}_{k=1}^n$, which satisfy, together with the operators of the family $={\left(B_j\right)}_{j=1}^n$, semilinear relations ${⅀}_i{f}_{ij}\left(\right)B_j{g}_{ij}\left(\right)=h\left(\right)$, ($f_{ij}$, $g_{ij}$, $h_j:{ℝ}^n\to ℂ$ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra $U_q^{\text{'}}\left(so\left(3\right)\right)$.

We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $𝒪$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $𝒪$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic...

We define sewn elliptic cohomologies for vertex algebras by sewing procedure for coboundary operators.